Daniel Gorenstein

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The Classification of the Finite Simple Groups

1994-11-18

Daniel Gorenstein , Richard Lyons , Ronald Solomon

General introduction to the special odd case General lemmas Theorem $C^*_2$: Stage 1 Theorem $C^*_2$: Stage 2 Theorem $C_2$: Stage 3 Theorem $C_2$: Stage 4 Theorem $C_2$: Stage 5 Theorem … General introduction to the special odd case General lemmas Theorem $C^*_2$: Stage 1 Theorem $C^*_2$: Stage 2 Theorem $C_2$: Stage 3 Theorem $C_2$: Stage 4 Theorem $C_2$: Stage 5 Theorem $C_3$: Stage 1 Theorem $C_3$: Stages 2 and 3 IV$_K$: Preliminary properties of $K$-groups Background references Expository references Glossary Index.

Finite Simple Groups

1982-01-01

Daniel Gorenstein

The local structure of finite groups of characteristic 2 type

1983-01-01

Daniel Gorenstein , Richard Lyons

Part I: Properties of $K$-groups and Preliminary Lemmas: Introduction Decorations of the known simple groups Local subgroups of the known simple groups Balance and signalizers Generational properties of $K$-groups Factorizations … Part I: Properties of $K$-groups and Preliminary Lemmas: Introduction Decorations of the known simple groups Local subgroups of the known simple groups Balance and signalizers Generational properties of $K$-groups Factorizations Miscellaneous general results and lemmas about $K$-groups Appendix by N. Burgoyne Part II: The Trichotomy Theorem: Odd standard form Signalizer functors and weak proper $2$-generated $p$-cores Almost strongly $p$-embedded maximal $2$-local subgroups References.

The characterization of finite groups with dihedral Sylow 2-subgroups. I

1965-03-01

Daniel Gorenstein , John H. Walter

The Classification of the Finite Simple Groups, Number 2

2002-11-15

Daniel Gorenstein , Richard Lyons , Ronald Solomon

The Classification of Finite Simple Groups

1983-01-01

Daniel Gorenstein

Finite groups with quasi-dihedral and wreathed Sylow 2-subgroups.

1970-01-01

J. L. Alperin , Richard Brauer , Daniel Gorenstein

The primary purpose of this paper is to give a complete classification of all finite simple groups with quasi-dihedral Sylow 2-subgroups. We shall prove that any such group must be … The primary purpose of this paper is to give a complete classification of all finite simple groups with quasi-dihedral Sylow 2-subgroups. We shall prove that any such group must be isomorphic to one of the groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L 3 left-parenthesis q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>L</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{L_3}(q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q identical-to minus 1 left-parenthesis mod 4 right-parenthesis comma upper U 3 left-parenthesis q right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≡</mml:mo> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mspace width="0.667em" /> <mml:mo stretchy="false">(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width="0.333em" /> <mml:mn>4</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mn>3</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>q</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">q \equiv - 1 \pmod 4,{U_3}(q)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q identical-to 1 left-parenthesis mod 4 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>≡</mml:mo> <mml:mn>1</mml:mn> <mml:mspace width="0.667em" /> <mml:mo stretchy="false">(</mml:mo> <mml:mi>mod</mml:mi> <mml:mspace width="0.333em" /> <mml:mn>4</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">q \equiv 1 \pmod 4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, or <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M 11"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>M</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>11</mml:mn> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{M_{11}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. We shall also carry out a major portion of the corresponding classification of simple groups with Sylow 2-subgroups isomorphic to the wreath product of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z Subscript 2 Sub Superscript n"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{Z_{{2^n}}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper Z 2 comma n greater-than-over-equals 2"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>Z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>≧</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">{Z_2},n \geqq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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Finite groups whose 2-subgroups are generated by at most 4 elements

1974-01-01

Daniel Gorenstein , Koichiro Harada

The Classification of the Finite Simple Groups, Number 4

1999-03-16

Daniel Gorenstein , Richard Lyons , Ronald Solomon

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On finite groups with dihedral Sylow 2-subgroups

1962-12-01

Daniel Gorenstein , John H. Walter

IntroductionMuch attention has recently been given to the characterization of classes of simple groups in terms of conditions which specify the centralizers of their revolutions or their Sylow 2-subgroups.(Cf.R. Brauer … IntroductionMuch attention has recently been given to the characterization of classes of simple groups in terms of conditions which specify the centralizers of their revolutions or their Sylow 2-subgroups.(Cf.R. Brauer [2]; R. Brauer, M.Suzuki, and G

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Finite Groups with Quasi-Dihedral and Wreathed Sylow 2-Subgroups

1970-09-01

J. L. Alperin , Richard Brauer , Daniel Gorenstein

An arithmetic theory of adjoint plane curves

1952-01-01

Daniel Gorenstein

For example, an adjoint curve to a curve with ordinary multiple points Pi, Ps, • • • , P, of multiplicities n, r2, ■ • ■ , r, respectively is … For example, an adjoint curve to a curve with ordinary multiple points Pi, Ps, • • • , P, of multiplicities n, r2, ■ • ■ , r, respectively is one which has an (/; -l)-fold point at each Pi.In this case, the degree of the fixed component is 23<_i r<(f<-1), while the number of conditions which the adjoint curves impose on the curves of sufficiently high degree is zZ'i^nin-D/l.

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The multiplicators of certain simple groups

1966-01-01

J. L. Alperin , Daniel Gorenstein

The recent paper of Steinberg [7] on the multiplicators of the finite simple groups of type, the classical determination of the multiplicators of the alternating groups by Schur [6], a … The recent paper of Steinberg [7] on the multiplicators of the finite simple groups of type, the classical determination of the multiplicators of the alternating groups by Schur [6], a similar result of Janko for his group [3] and the (unpublished) work of J. G. Thompson on the Mathieu groups cover all but three families of known simple groups. In this paper we give a simple determination of the multiplicators for two of these families, namely the Suzuki groups and the Ree groups of characteristic three. Our results are well known for the Suzuki groups, with the exception of the one of smallest order, while the determination for the Ree groups of characteristic three has been accomplished by J. H. Walter with the use of some deep theorems of modular character theory. However, our main tool is an elementary lemma of Lie type involving a very crude numerical estimate. In addition, we calculate the multiplicator for the smallest Suzuki group. Furthermore, preliminary investigations indicate that our methods might show that the multiplicators, or at least their 2-primary components, are trivial for the remaining family of Ree groups of characteristic two defined over GF(22n+') for all n > N, where N is fairly small. Our lemma deals with a single automorphism; a suitable generalization of this result to a pair of commuting automorphisms would suffice to prove the preceding statement. Our main results are:

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The characterization of finite groups with dihedral Sylow 2-subgroups—II

1965-06-01

Daniel Gorenstein , John H. Walter

The characterization of finite groups with dihedral Sylow 2-subgroups. III

1965-09-01

Daniel Gorenstein , John H. Walter

Centralizers of involutions in balanced groups

1972-02-01

Daniel Gorenstein , John H. Walter

Finite simple groups and their classification

1974-03-01

Daniel Gorenstein

The classification of finite simple groups I. Simple groups and local analysis

1979-01-01

Daniel Gorenstein

DedicationIn memory of Richard Brauer, for his pioneering studies of finite simple groups.It is indeed unfortunate that Richard Brauer did not live to see the complete classification of the finite … DedicationIn memory of Richard Brauer, for his pioneering studies of finite simple groups.It is indeed unfortunate that Richard Brauer did not live to see the complete classification of the finite simple groups.He had devoted the past thirty years largely to their study and it is difficult to overestimate the impact he made on the subject.Early on, he realized the intimate relationship between the structure of a group and the centralizers of its involutions (elements of order 2).He established both qualitative and quantitative connections.As an example of the first, he showed by an elementary argument that there are at most a finite number of simple groups with a specified centralizer of an involution [30]; and of the second, he proved that if the centralizer of an involution in a simple group G is isomorphic to the general linear group GL(2, q) over the finite field with q elements, q odd, then either G is isomorphic to the three-dimensional projective special linear group L 3 (q), 2 or else q = 3 and G is isomorphic to the smallest Mathieu group M n of order 8 • 9 • 10 • 11 [27], [28].This last result, which Brauer announced in his address at the International Congress of Mathematicians in Amsterdam in 1954, represented the starting point for the classification of simple groups in terms of the structure of the centralizers of their involutions.Moreover, it foreshadowed the basic fact that conclusions of general classification theorems would necessarily include sporadic simple groups as exceptional cases (M n being the sporadic group of least order).The methods which Brauer used were almost entirely representationtheoretic and character-theoretic.In the middle 1930s he introduced and developed the concept of modular characters of a finite group.He soon realized the power of these ideas, which played an instrumental role in his proof of the Artin conjecture on Ê-series in algebraic number fields.Likewise he saw that these techniques provided a powerful tool for investigating simple groups.From the middle 1940s until his death, Brauer systematically devel-

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Balance and generation in finite groups

1975-02-01

Daniel Gorenstein , John H. Walter

Finite groups with Sylow 2-subgroups of class two. I

1975-01-01

Robert H. Gilman , Daniel Gorenstein

In this paper we classify finite simple groups whose Sylow <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-subgroups have nilpotency class two. In this paper we classify finite simple groups whose Sylow <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-subgroups have nilpotency class two.

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Finite Groups Admitting a Fixed-Point-Free Automorphism of Order 4

1961-01-01

Daniel Gorenstein , I. N. Herstein

Recently, in a remarkable piece of work [4, 5] John Thompson has proved a result which implies as an immediate corollary the well-known Frobenius conjecture, namely that a finite group … Recently, in a remarkable piece of work [4, 5] John Thompson has proved a result which implies as an immediate corollary the well-known Frobenius conjecture, namely that a finite group admitting a fixed-point-free automorphism (i. e., leaving only the identity element fixed) of prime order must be nilpotent. However, non-nilpotent groups are known which admit fixedpoint-free automorphisms of composite order. In all these cases one notices that the groups in question are solvable. Although the sample is rather restricted, it is not too unnatural to ask whether the condition that a finite group admit such an automorphism is strong enough to force solvability of the group. This question is related to another problem, which seems. equally difficult, which asks whether a finite group containing a cyclic subgroup which is its own normalizer must be composite. In the present paper we shall prove that a group G possessiilg a fixedpoint-f ree automorphism of order 4 is solvable. Although many of the ideas used carry over to the case in which 4 has order pq, and especially 2q, our key lemmas use the fact that 4 has order 4 in a crucial way. The proof depends upon a theorem of Philip Ilall which asserts that a finite group G is solvable if for every factorization of o(G) into relatively prime numbers m and n, G contains a subgroup of order m. We show (Lemma 7) that a group G which has a fixed-point-free automorphism of order 4 satisfies the conditions of H all's theorem. Once we k-now that G is solvable it is not difficult to prove that its commutator subgroup is nilpotent (Theorem 2). This -fact was also observed by Thompson. Graham Higman has shown [3] that there is a bound to the class of a p-group P which possesses an automorphism p of prime order q without fixedpoints. This does not carry over to automorphisms of composite order, for at the end of the paper we give an example due to Thompson of a family of p-groups of arbitrary high class each of which admits a fixed-point-free automorphism of order 4.

The Classification of the Finite Simple Groups, Number 9

2021-02-22

Inna Capdeboscq , Daniel Gorenstein , Richard Lyons +1 more

Finite Groups the Centralizers of whose involutions have Normal 2-Complements

1969-01-01

Daniel Gorenstein

In this paper we shall classify all finite groups in which the centralizer of every involution has a normal 2-complement. For brevity, we call such a group an I-group. To … In this paper we shall classify all finite groups in which the centralizer of every involution has a normal 2-complement. For brevity, we call such a group an I-group. To state our classification theorem precisely, we need a preliminary definition. As is well-known, the automorphism group G = PΓL(2, q ) of H = PSL(2, q ), q = p n , is of the form G = LF , where L = PGL(2, q ), L ⨞ G, F is cyclic of order n , L ∩ F = 1, and the elements of F are induced from semilinear transformations of the natural vector space on which GL(2, q ) acts; cf. ( 3 , Lemma 2.1) or ( 7 , Lemma 3.3). It follows at once ( 4 , Lemma 2.1; 8 , Lemma 3.1) that the groups H and L are each I-groups. Moreover, when q is an odd square, there is another subgroup of G in addition to L that contains H as a subgroup of index 2 and which is an I-group.

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On almost-commuting permutations

1962-09-01

Daniel Gorenstein , Reuben Sandler , W. H. Mills

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On finite groups with Sylow 2-subgroups of typeA n ,n=8, 9, 10, 11

1970-01-01

Daniel Gorenstein , Koichiro Harada

Finite simple groups of low 2-rank and the families $G_2 \left( q \right),\,D_4^2 \left( q \right),\,q$ odd

1971-11-01

Daniel Gorenstein , Koichiro Harada

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The Classification of the Finite Simple Groups, Number 10

2023-10-05

Inna Capdeboscq , Daniel Gorenstein , Richard Lyons +1 more

The Classification of the Finite Simple Groups, Number 6

2004-12-07

Daniel Gorenstein , Richard Lyons , Ronald Solomon

Nonsolvable finite groups with solvable 2-local subgroups

1976-02-01

Daniel Gorenstein , Richard Lyons

Classifying the finite simple groups

1986-01-01

Daniel Gorenstein

CHAPTER IV: Outline of a Unitary Proof 1.The over-all plan 2. Groups of even type 3. Small groups 4. Uniqueness subgroups 5.The sets Se p {G) 6. Generic and nongeneric … CHAPTER IV: Outline of a Unitary Proof 1.The over-all plan 2. Groups of even type 3. Small groups 4. Uniqueness subgroups 5.The sets Se p {G) 6. Generic and nongeneric simple groups 7. The major case division 8.The working hypothesis 9. fc-balance and signalizer functors 10.Internally eliminating Rad(G) 11.Brauer form 12. Strong Brauer form 13. Internal definition of Soc( G ) 14.The identification of G 15. Concluding comments

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On finite simple groups of characteristic 2 type

1969-01-01

Daniel Gorenstein

The $\pi$-layer of a finite group

1971-12-01

Daniel Gorenstein , John H. Walter

A fundamental property of a r-separable group H, r a set of primes, is the fact that H is v-constrained; that is, O(H/O,(H) contains its own centralizer in H/O,(H) (Theorem … A fundamental property of a r-separable group H, r a set of primes, is the fact that H is v-constrained; that is, O(H/O,(H) contains its own centralizer in H/O,(H) (Theorem 6.3.2 of [3]').This concept plays a basic role in al- most all of the general classification problems solved to date (see, for example, [1], [2], [4], [5], [6], [7], [8]).On the other hand, in the classification of groups with abelian Sylow 2-subgroups [9], large portions of this analysis involve non-constrained subgroups of the simple group under consideration.It is clear that the latter situation will be typical in more general classification problems.It is therefore natural to ask whether there exists in an arbitrary finite group H a suitably chosen subgroup of H/O,(H) containing its own centra- lizer which can be used effectively in place of O(H/O,(H)).In Theorem 1, we prove the existence of such a subgroup.Then in Theorem 2, we study the inverse image of this subgroup in H.This leads to the concept of the --layer and the r-components of the group H.In Section 3 we derive a number of elementary properties of the r-layer and the r-components of a group; and in Section 4, we prove some general lemmas which are useful in analyzing the r-components of a given group.It will be convenient to adopt what we may call the "bar" convention: Namely, if H is a group and X is an element, subset, or subgroup of H and if//is a homomorphic image of H, then 2: will always denote the image of X intl.2. The --layer of a group We say that a group G is quasisimple if it possesses a perfect normal sub- group H with the following properties" (i) H/Z(H) is a nonabelian simple group, and (ii) Ca(H) _ Z(H).These conditions imply that H is the unique such subgroup of G and hence that H is characteristic in G. Further- more, since H is perfect, no proper subgroup of H covers H/Z(H).If G H, we say that G is perfect quasisimple.A central product L of perfect quasisimple groups L, 1 _< i _< r, will be said to be semisimple.(This definition is an extension of the usual notion of semisimple.)The factors L of L are actually uniquely determined.In- deed, our conditions imply that Z(L) II=l Z(L), whence L L/Z (L)

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On the centralizers of involutions in finite groups

1969-02-01

Daniel Gorenstein

On finite groups with Sylow 2-subgroups of type Ân, n = 8, 9, 10, and 11

1971-10-01

Daniel Gorenstein , Koichiro Harada

Finite Groups in Which Sylow 2-Subgroups are Abelian and Centralizers of Involutions are Solvable

1965-01-01

Daniel Gorenstein

The purpose of this paper is to establish the following theorem : Theorem 1. Let be a finite group with abelian Sylow 2-subgroups in which the centralizer of every involution … The purpose of this paper is to establish the following theorem : Theorem 1. Let be a finite group with abelian Sylow 2-subgroups in which the centralizer of every involution is solvable. Then either is solvable or else /O is isomorphic to a subgroup of PΓL(2, q) containing PSL(2, q), where either q = 3 or 5 ( mod 8), q ≥ 5, or q = 2 n , n ≥ 2.

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The Classification of the Finite Simple Groups, Number 5

2002-09-10

Daniel Gorenstein , Richard Lyons , Ronald Solomon

Finite Groups Whose Sylow 2-Subgroups are the Direct Product of Two Dihedral Groups

1972-01-01

Daniel Gorenstein , Koichiro Harada

A characterization of the Higman-Sims simple group

1973-03-01

Daniel Gorenstein , Morton E. Harris

On a theorem of Philip Hall

1966-10-01

Daniel Gorenstein

Lemma 8.2 of ''Solvability of Groups of Odd Order" by W. Feit and J Lemma 8.2 of ''Solvability of Groups of Odd Order" by W. Feit and J

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Finite groups with Sylow 2-subgroups of type $PS_p (4,q),q$ odd

1973-10-25

Daniel Gorenstein , Koichiro Harada

On the maximal subgroups of finite simple groups

1964-07-01

Daniel Gorenstein , John H. Walter

An Arithmetic Theory of Adjoint Plane Curves

1952-05-01

Daniel Gorenstein

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A vanishing theorem for cohomology

1972-01-01

J. L. Alperin , Daniel Gorenstein

A criterion is given for ${H^0}(G,A) = {H^1}(G,A) = 0$, where A criterion is given for ${H^0}(G,A) = {H^1}(G,A) = 0$, where

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A Class of Solvable Groups

1959-01-01

Daniel Gorenstein , I. N. Herstein

Numerous studies have been made of groups, especially of finite groups, G which have a representation in the form AB , where A and B are subgroups of G . … Numerous studies have been made of groups, especially of finite groups, G which have a representation in the form AB , where A and B are subgroups of G . The form of these results is to determine various grouptheoretic properties of G, for example, solvability, from other group-theoretic properties of the subgroups A and B . More recently the structure of finite groups G which have a representation in the form ABA , where A and B are subgroups of G , has been investigated. In an unpublished paper, Herstein and Kaplansky (2) have shown that if A and B are both cyclic, and at least one of them is of prime order, then G is solvable. Also Gorenstein (1) has completely characterized ABA groups in which every element is either in A or has a unique representation in the form aba' , where a, a’ are in A , and b ≠ 1 is in B .

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Triply transitive groups in which only the identity fixes four letters

1961-09-01

Daniel Gorenstein , Daniel Hughes

Doubly transitive groups in which only the identity fixes three letters have been the subject of recent investigations by Feit [1] and Suzuki [3], [4].In the present paper we shall … Doubly transitive groups in which only the identity fixes three letters have been the subject of recent investigations by Feit [1] and Suzuki [3], [4].In the present paper we shall consider the corresponding class of triply transitive groups--that is, triply transitive groups in which only the identity fixes four letters--and shall prove the following:THEOREM.If G3 is a finite triply transitive group in which only the identity fixes 4 letters, then G3 is one of the following:(a) sharply 4-fold transitive, and hence is either the Mathieu group Mll, the symmetric group S or $5 or the alternating group A6 (b) sharply triply transitive, and hence is either the linear fractional group Lq over some GF(q) or the group Lrq of transformationsx-(a () + b)/(cx () + d), over some GF(q), q odd, where A ad bc O, and x if A is a square, (r (A)" x --\x if A is a nonsquare;(c) the full semilinear fractional group Pq of all transformationsx --(ax"b)/(cx" -d) over some GF(2q) where q is a prime, ad bc O, and a is an automorphism of GF(2q).Our proof relies heavily on the work of Felt and Suzuki as well as on some earlier results of Zassenhaus [6].In Section 1 we list for the sake of clarity most of the known results which we shall need.Section 2 is devoted to an initial reduction of the theorem.In Sections 3 and 4, respectively, we then treat the cases that Ga is of odd degree and even degree.1. Summary of known resultsIf G is a t-fold transitive group of degree n -k 1 on the letters P1, P2 P,-I QI Q2, Qt we shall denote by G the subgroup of Gt fixing Q+,Q+.,...,Qt for i 0, 1,2,...,t-1.If no permutation

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Finite Groups of Sectional 2-Rank at Most 4

1973-01-01

Daniel Gorenstein , Koichiro Harada

The flatness of signalizer functors on finite groups

1969-12-01

Daniel Gorenstein

Transfer and fusion in finite groups

1967-06-01

J. L. Alperin , Daniel Gorenstein

Finite Groups With Product Fusion

1975-01-01

Daniel Gorenstein , Morton E. Harris

In several recent papers concerning the classification of finite simple groups, the investigators were forced to treat subsidiary problems involving a group G whose Sylow 2-subgroup S is a direct … In several recent papers concerning the classification of finite simple groups, the investigators were forced to treat subsidiary problems involving a group G whose Sylow 2-subgroup S is a direct product S = S, x S2 in which the fusion of 2-elements of G corresponds to that of the direct product of two groups having S, and S2, respectively, as their Sylow 2-subgroups. For example, in the study of groups, with Sylow 2-subgroups of type G2(q) or Psp(4, q), q odd, the determination of the structure of the centralizer of a central involution requires a solution of this subsidiary problem in the case in which S, and S2 are dihedral groups ([9]). The analogous problem arises with S, dihedral and S2 wreathed in Mason's work on groups with Sylow 2-subgroups of type L4(q), q odd. Likewise in the analysis of groups with Sylow 2-subgroups of type Psp(6, q), q odd, two other such direct product problems arise in the course of determining the structure of the centralizers of involutions. It would therefore appear to be quite useful to have available an effective result about the structure of such a group G when the direct factors S, S2 of S are arbitrary. It is the object of this paper to establish such a general result. Suppose that we are trying to determine all simple groups G satisfying some set of conditions. In such a problem, it is very likely that one can reduce to the case in which the critical composition factors of the proper subgroups of G are of known type. Moreover, in those cases in which such a classification theorem has been obtained, specific properties of such composition factors have entered into the arguments. It is therefore reasonable and most likely necessary to impose some general conditions on appropriate composition factors of the proper subgroups of the group G under investigation. It turns out that a single assumption stated in terms of the notion of balance is all that is necessary. Moreover, we make this assumption only on the composition factors of certain subgroups of G which possess a Sylow 2-subgroup of the form T1 x T2with T, z Si, i = 1, 2.

Finite groups with Sylow 2-subgroups of class two. II

1975-01-01

Robert H. Gilman , Daniel Gorenstein

In this paper we classify finite simple groups whose Sylow <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-subgroups have nilpotence class two. In this paper we classify finite simple groups whose Sylow <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-subgroups have nilpotence class two.

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Theorem 𝒞₄*: Stage A3. 𝒦ℳ-singularities

2023-10-05

Inna Capdeboscq , Daniel Gorenstein , Richard Lyons +1 more

The Classification of the Finite Simple Groups, Number 10

2023-10-05

Inna Capdeboscq , Daniel Gorenstein , Richard Lyons +1 more

General group-theoretic lemmas

2023-10-05

Inna Capdeboscq , Daniel Gorenstein , Richard Lyons +1 more

Properties of 𝒦-groups

2023-10-05

Inna Capdeboscq , Daniel Gorenstein , Richard Lyons +1 more

The Classification of the Finite Simple Groups, Number 9

2021-02-22

Inna Capdeboscq , Daniel Gorenstein , Richard Lyons +1 more

General group-theoretic lemmas, and recognition theorems

2021-02-22

Inna Capdeboscq , Daniel Gorenstein , Richard Lyons +1 more

Preliminary properties of 𝒦-groups

2021-02-22

Inna Capdeboscq , Daniel Gorenstein , Richard Lyons +1 more

Introduction to theorem 𝒞₅

2021-02-22

Inna Capdeboscq , Daniel Gorenstein , Richard Lyons +1 more

The Classification of the Finite Simple Groups, Number 8

2018-12-12

Daniel Gorenstein , Richard Lyons , Ronald Solomon

Theorem 𝒞*₇: Stage 4b+—A large Lie-type subgroup 𝒢₀ for 𝓅>2

2018-12-12

Daniel Gorenstein , Richard K. Lyons , Ronald Solomon

Preliminary properties of 𝒦-groups

2018-12-12

Daniel Gorenstein , Richard K. Lyons , Ronald Solomon

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Theorem 𝒞*₇: Stage 4b+—A large Lie-type subgroup 𝒢₀ for 𝓅=2

2018-12-12

Daniel Gorenstein , Richard Lyons , Ronald Solomon

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The Classification of the Finite Simple Groups, Number 7

2018-02-15

Daniel Gorenstein , Richard Lyons , Ronald Solomon

General group-theoretic lemmas

2018-02-15

Daniel Gorenstein , Richard K. Lyons , Ronald Solomon

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Properties of 𝒦-groups

2018-02-15

Daniel Gorenstein , Richard Lyons , Ronald Solomon

Theorem 𝒞*₇: Stage 3b

2018-02-15

Daniel Gorenstein , Richard K. Lyons , Ronald Solomon

The Classification of the Finite Simple Groups, Number 6

2004-12-07

Daniel Gorenstein , Richard Lyons , Ronald Solomon

𝐼𝑉_{𝐾}: Preliminary properties of 𝒦-groups

2004-12-07

Daniel Gorenstein , Richard Lyons , Ronald Solomon

General lemmas

2004-12-07

Daniel Gorenstein , Richard K. Lyons , Ronald Solomon

The Classification of the Finite Simple Groups, Number 2

2002-11-15

Daniel Gorenstein , Richard Lyons , Ronald Solomon

The Classification of the Finite Simple Groups, Number 5

2002-09-10

Daniel Gorenstein , Richard Lyons , Ronald Solomon

General group-theoretic results

2002-09-10

Daniel Gorenstein , Richard Lyons , Ronald Solomon

Properties of 𝒦-Groups

2002-09-10

Daniel Gorenstein , Richard K. Lyons , Ronald Solomon

Properties of 𝒦-groups

1999-03-16

Daniel Gorenstein , Richard Lyons , Ronald Solomon

Strongly embedded subgroups and related conditions on involutions

1999-03-16

Daniel Gorenstein , Richard Lyons , Ronald Solomon

The Classification of the Finite Simple Groups, Number 4

1999-03-16

Daniel Gorenstein , Richard Lyons , Ronald Solomon

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General lemmas

1999-03-16

Daniel Gorenstein , Richard K. Lyons , Ronald Solomon

The alternating groups and the twenty-six sporadic groups

1997-12-02

Daniel Gorenstein , Richard Lyons , Ronald Solomon

General properties of 𝒦-groups

1997-12-02

Daniel Gorenstein , Richard Lyons , Ronald Solomon

Local subgroups of groups of Lie type, I

1997-12-02

Daniel Gorenstein , Richard Lyons , Ronald Solomon

Some theory of linear algebraic groups

1997-12-02

Daniel Gorenstein , Richard Lyons , Ronald Solomon

Coverings and embeddings of quasisimple 𝒦-groups

1997-12-02

Daniel Gorenstein , Richard Lyons , Ronald Solomon

The finite groups of Lie type

1997-12-02

Daniel Gorenstein , Richard Lyons , Ronald Solomon

Local subgroups of groups of Lie type, II

1997-12-02

Daniel Gorenstein , Richard K. Lyons , Ronald Solomon

The Classification of the Finite Simple Groups

1994-11-18

Daniel Gorenstein , Richard Lyons , Ronald Solomon

On aschbacher’s localC(G; T) theorem

1993-06-01

Daniel Gorenstein , Richard Lyons

On Split (B, N)-Pairs of Rank 2

1993-05-01

Daniel Gorenstein , Richard Lyons , Ron Solomon

A Brief History of the Sporadic Simple Groups

1993-01-01

Daniel Gorenstein

In these lectures I shall give a brief historical summary of the 26 sporadic simple groups. Apart from the Mathieu groups, their discovery was closely intertwined with theoretical developments in … In these lectures I shall give a brief historical summary of the 26 sporadic simple groups. Apart from the Mathieu groups, their discovery was closely intertwined with theoretical developments in finite simple group theory from 1950–1980, and the historical perspective provides an understanding of key aspects of the classification of the finite simple groups. The material to be covered here is primarily an extraction from a fuller discussion of the known simple groups that appears in Chapter 2 of my book, Finite Simple Groups, Plenum (New York), 1982.

Nonsolvable signalizer functors revisited

1990-09-01

Daniel Gorenstein , Richard Lyons

Signalizer functors and fusion in finite groups

1990-05-01

Daniel Gorenstein , Richard Lyons

Classifying the finite simple groups

1986-01-01

Daniel Gorenstein

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The Classification of Groups of Component Type

1983-01-01

Daniel Gorenstein

Simple Groups of 2-Rank ≤ 2

1983-01-01

Daniel Gorenstein

The local structure of finite groups of characteristic 2 type

1983-01-01

Daniel Gorenstein , Richard Lyons

The B-Theorem and Locally Unbalanced Groups

1983-01-01

Daniel Gorenstein

The Classification of Finite Simple Groups

1983-01-01

Daniel Gorenstein

Groups of noncharacteristic 2 type

1983-01-01

Daniel Gorenstein

Never before in the history of mathematics has there been an individual theorem whose proof has required 10,000 journal pages of closely reasoned argument. Who could read such a proof, … Never before in the history of mathematics has there been an individual theorem whose proof has required 10,000 journal pages of closely reasoned argument. Who could read such a proof, let alone communicate it to others? But the classification of all finite simple groups is such a theorem-its complete proof, developed over a 30-year period by about 100 group theorists, is the union of some 500 journal articles covering approximately 10,000 printed pages. How then is one who has lived through it all to convey the richness and variety of this monumental achievement? Yet such an attempt must be made, for without the existence of a coherent exposition of the total proof, there is a very real danger that it will gradually become lost to the living world of mathematics, buried within the dusty pages of forgotten journals. For it is almost impossible for the uninitiated to find the way through the tangled proof without an experienced guide; even the 500 papers themselves require careful selection from among some 2,000 articles on simple group theory, which together include often attractive byways, but which serve only to delay the journey.

Centralizers of Involutions in Simple Groups

1983-01-01

Daniel Gorenstein

Signalizer functors, proper 2-generated cores, and nonconnected groups

1982-03-01

Daniel Gorenstein , Richard Lyons

Finite Simple Groups

1982-01-01

Daniel Gorenstein

Coauthor	Papers Together
Richard Lyons	37
Ronald Solomon	35
Richard K. Lyons	9
John H. Walter	8
Inna Capdeboscq	8
Koichiro Harada	7
J. L. Alperin	5
Robert H. Gilman	4
Richard Brauer	3
I. N. Herstein	3
J. L. Alperin	3
I. N. Herstein	3
Michael Aschbacher	2
Morton E. Harris	2
Ron Solomon	1
Reuben Sandler	1
Michael Artin	1
Daniel Hughes	1
W. H. Mills	1
Maxwell Rosenlicht	1

On the centralizers of involutions in finite groups

1969-02-01

Daniel Gorenstein

Solvable signalizer functors on finite groups

1972-04-01

David M. Goldschmidt

The Characterization of Finite Groups with Abelian Sylow 2-Subgroups

1969-05-01

John H. Walter

Central elements in core-free groups

1966-11-01

George Glauberman

Nonsolvable finite groups all of whose local subgroups are solvable

1968-01-01

John G. Thompson

Notation and definitions 384 3. Statement of main theorem and corollaries 388 4. Proofs of corollaries 389 5. Preliminary lemmas 389 5.1.Inequalities and modules 389 5.2.7r-reducibility and fl,(®) 395 5.3.Groups … Notation and definitions 384 3. Statement of main theorem and corollaries 388 4. Proofs of corollaries 389 5. Preliminary lemmas 389 5.1.Inequalities and modules 389 5.2.7r-reducibility and fl,(®) 395 5.3.Groups of symplectic type 397 5.4.^-groups, ^-solvability and F((&) 400 5.5.Groups of low order 404 5.6.2-groups, involutions and 2-length 409 5.7.Factorizations 423 5.8.Miscellaneous 426 6.A transitivity theorem 428 i 9 68]

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A new finite simple group with abelian Sylow 2-subgroups and its characterization

1966-03-01

Zvonimir Janko

Centralizers of involutions in balanced groups

1972-02-01

Daniel Gorenstein , John H. Walter

The flatness of signalizer functors on finite groups

1969-12-01

Daniel Gorenstein

2-Signalizer functors on finite groups

1972-05-01

David M. Goldschmidt

The characterization of finite groups with dihedral Sylow 2-subgroups. I

1965-03-01

Daniel Gorenstein , John H. Walter

The $\pi$-layer of a finite group

1971-12-01

Daniel Gorenstein , John H. Walter

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On the maximal subgroups of finite simple groups

1964-07-01

Daniel Gorenstein , John H. Walter

Transitive gruppen gerader ordnung, in denen jede involution genau einen punkt festläβt

1971-04-01

Helmut Bender

Balance and generation in finite groups

1975-02-01

Daniel Gorenstein , John H. Walter

A Characteristic Subgroup of a p-Stable Group

1968-01-01

George Glauberman

Let p be a prime, and let S be a Sylow p -subgroup of a finite group G . J. Thompson ( 13 ; 14 ) has introduced a characteristic … Let p be a prime, and let S be a Sylow p -subgroup of a finite group G . J. Thompson ( 13 ; 14 ) has introduced a characteristic subgroup J R (S) and has proved the following results: (1.1) Suppose that p is odd. Then G has a normal p-complement if and only if C(Z(S)) and N(J R (S)) have normal p-complements .

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On 2-groups with no normal abelian subgroups of rank 3, and their occurrence as Sylow 2-subgroups of finite simple groups

1970-01-01

Anne R. MacWilliams

We prove that in a finite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-group with no normal Abelian subgroup of rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="greater-than-over-equals … We prove that in a finite <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-group with no normal Abelian subgroup of rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="greater-than-over-equals 3"> <mml:semantics> <mml:mrow> <mml:mo>≧</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\geqq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, every subgroup can be generated by four elements. This result is then used to determine which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with no normal Abelian subgroup of rank <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="greater-than-over-equals 3"> <mml:semantics> <mml:mrow> <mml:mo>≧</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\geqq 3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> can occur as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S 2"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>S</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{S_2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>’s of finite simple groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, under certain assumptions on the embedding of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T"> <mml:semantics> <mml:mi>T</mml:mi> <mml:annotation encoding="application/x-tex">T</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

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Finite groups with quasi-dihedral and wreathed Sylow 2-subgroups.

1970-01-01

J. L. Alperin , Richard Brauer , Daniel Gorenstein

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Chapter I, from Solvability of groups of odd order, Pacific J. Math, vol. 13, no. 3 (1963

1963-09-01

Walter Feit , John Thompson

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Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen.

1911-01-01

Julia Schur

Article Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen. was published on January 1, 1911 in the journal Journal für die reine und angewandte Mathematik … Article Über die Darstellung der symmetrischen und der alternierenden Gruppe durch gebrochene lineare Substitutionen. was published on January 1, 1911 in the journal Journal für die reine und angewandte Mathematik (volume 1911, issue 139).

ON FINITE GROUPS OF EVEN ORDER WHOSE 2-SYLOW GROUP IS A QUATERNION GROUP

1959-12-01

Richard Brauer , Michio Suzuki

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The local structure of finite groups of characteristic 2 type

1983-01-01

Daniel Gorenstein , Richard Lyons

On Solvable Signalizer Functors in Finite Groups

1976-07-01

George Glauberman

On the p -Length of p -Soluble Groups and Reduction Theorems for Burnside's Problem

1956-01-01

P. Hall , Graham Higman

Finite Groups Admitting a Fixed-Point-Free Automorphism of Order 4

1961-01-01

Daniel Gorenstein , I. N. Herstein

Sylow intersections and fusion

1967-06-01

J. L. Alperin

On groups with abelian Sylow 2-subgroups

1970-01-01

Helmut Bender

A characterization of the one-dimensional unimodular projective groups over finite fields

1958-12-02

Richard Brauer , Michio Suzuki , G. E. Wall

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A Characterization of the finite simple group U4(3)

1969-08-01

Kok-Wee Phan

The aim of this paper is to give a characterization of the finite simple group U 4 (3) i.e. the 4-dimensional projective special unitary group over the field of 9 … The aim of this paper is to give a characterization of the finite simple group U 4 (3) i.e. the 4-dimensional projective special unitary group over the field of 9 elements. More precisely, we shall prove the following result.

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On finite groups with dihedral Sylow 2-subgroups

1962-12-01

Daniel Gorenstein , John H. Walter

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A conjugation family for finite groups

1970-09-01

David M. Goldschmidt

Involutions in Chevalley groups over fields of even order

1976-10-01

Michael Aschbacher , Gary M. Seitz

Let G = G(q) be a Chevalley group defined over a field F q of characteristic 2. In this paper we determine the conjugacy classes of involutions in Aut( G … Let G = G(q) be a Chevalley group defined over a field F q of characteristic 2. In this paper we determine the conjugacy classes of involutions in Aut( G ) and the centralizers of these involutions. This study was begun in the context of a different problem.

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Groups with a (B, N)-pair of rank 2. I

1973-03-01

Paul Fong , Gary M. Seitz

A Family of Simple Groups Associated with the Simple Lie Algebra of Type (G 2 )

1961-07-01

Rimhak Ree

On finite simple groups of characteristic 2 type

1969-01-01

Daniel Gorenstein

On Ree's Series of Simple Groups

1966-01-01

Harold N. Ward

Ree recently discovered a series of finite simple groups related to the simple Lie algebra of type (G2) [5; 6].We have determined the irreducible characters of these groups.In this work, … Ree recently discovered a series of finite simple groups related to the simple Lie algebra of type (G2) [5; 6].We have determined the irreducible characters of these groups.In this work, we do not use the actual definition of Ree's groups, but only the properties (l)-( 5) given below.Since these are sufficient to determine the bulk of the character tables of these groups, it is our hope that they actually characterize the groups completely.The five properties used are (1) If G is one of these groups, the order of G is even and the 2-Sylow subgroup of G is elementary Abelian of order 8.(2) G has no normal subgroup of index 2.(3) There is an involution / (an element of order 2) in G such that if C(J) is the centralizer of / in G and (J) is the group generated by /, then C(J)/(J) is isomorphic to LF(2, q), the linear fractional group in two variables over a field of q elements.Here we restrict q by q^3 (mod 8) and q^27.It follows that C(J) is the direct product of (ƒ) and a subgroup F of G which is isomorphic to LF(2, q).In F there is an element R of order (q -l)/2 f and we also have (4) If R a 5*l, then C(R a )QC(J).Finally, there is an element S of F with order (q+l)/2.Let J' = S* where /=(g+l)/4.Then S 2 generates the commutator subgroup of C(J, J'), the centralizer of (J, J').We have (5) ForS 2 , C(S*)QC(J).This condition can actually be replaced by the weaker condition (5*) Let A €N(J, J"'), the normalizer in G of (ƒ, /'>, but A $ C(J, J 7 )-Let A z = 1.Then A does not commute with S 2 .From these conditions on G we derive a number of results.The approach is almost entirely by means of characters, both ordinary and modular.Condition (4) leads to two families of exceptional characters related to the classes of R a ( 5^1) and JR a (T^J).These characters are the characters of 2-defect 1. Results of Brauer [l; 2; 3] then lead to two possibilities for the principal 2-block (the 2-block containing the character which is 1 everywhere) ; one contains seven

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Finite groups whose 2-subgroups are generated by at most 4 elements

1974-01-01

Daniel Gorenstein , Koichiro Harada

Finite Groups With a Proper 2-Generated Core

1974-10-01

Michael Aschbacher

Finite Groups Whose Sylow 2-Subgroups are the Direct Product of Two Dihedral Groups

1972-01-01

Daniel Gorenstein , Koichiro Harada

Zur Darstellungstheorie der Gruppen endlicher Ordnung. II

1959-12-01

Richard Brauer

On finite groups containing an element of order four which commutes only with its powers

1959-06-01

Michio Suzuki

This paper is a study of a particular case of the following question" Let K be a subset of a finite group/-/.Suppose that another finite group contains H in such … This paper is a study of a particular case of the following question" Let K be a subset of a finite group/-/.Suppose that another finite group contains H in such a way that the centralizer in G of any element of K is con- tained in H. Then what can we say about the structure of G? In particular for given H and K, are there infinitely many simple groups G satisfying the

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2-Fusion in Finite Groups

1974-01-01

David M. Goldschmidt

Suppose A c T c G are groups such that whenever a E A, g e G, and ag E T, then as e A. In this situation, we say … Suppose A c T c G are groups such that whenever a E A, g e G, and ag E T, then as e A. In this situation, we say that A is strongly closed in T with respect to G. We are concerned here with the case where G is finite, T is a Sylow 2-subgroup of G, and A is Abelian. In this case, we call G an S(A)-group. Our objective is to determine the composition factors of AG, the normal closure of A in G, when G is an S(A)-group. For this purpose, we make the following definitions: Suppose L is a perfect group such that L/Z(L) is simple. (Such a group is called quasi-simple.) We shall say that L is of type I provided L/Z(L) is isomorphic to one of the following: (a) L2(2), n > 3. (b) Sz(22n+l), n~ > 1. (c) U3 (2 ), n > 2. We shall say that L is of type II provided Z(L) has odd order and L = L/Z(L) satisfies one of the following conditions: (d) L L2(q), q 3, 5 (mod 8). (e) L is simple and contains an involution t such that (1) I L: CL(t) I is odd, (2) CL(t) = x K, (3) K contains a normal subgroup E of odd index such that CK(E) = 1 and E L2(q)g q _ 3, 5 (mod 8).

On finite groups with Sylow 2-subgroups of typeA n ,n=8, 9, 10, 11

1970-01-01

Daniel Gorenstein , Koichiro Harada

Near solvable signalizer functors on finite groups

1982-09-01

Patrick Paschal McBride

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Beziehungen zwischen den Fixpunktzahlen von Automorphismengruppen einer endlichen Gruppe

1960-04-01

Helmut Wielandt

A Class of Solvable Groups

1959-01-01

Daniel Gorenstein , I. N. Herstein

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Finite groups with a proper 2-generated core

1974-01-01

Michael Aschbacher

H. Bender’s classification of finite groups with a strongly embedded subgroup is an important tool in the study of finite simple groups. This paper proves two theorems which classify finite … H. Bender’s classification of finite groups with a strongly embedded subgroup is an important tool in the study of finite simple groups. This paper proves two theorems which classify finite groups containing subgroups with similar but somewhat weaker embedding properties. The first theorem, classifying the groups of the title, is useful in connection with signalizer functor theory. The second theorem classifies a certain subclass of the class of finite groups possessing a permutation representation in which some involution fixes a unique point.

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