This is a general survey of the Classification of the Finite Simple Groups with particular emphasis on the current project of Gorenstein, Lyons and Solomon (GLS) directed towards the revision โฆ
This is a general survey of the Classification of the Finite Simple Groups with particular emphasis on the current project of Gorenstein, Lyons and Solomon (GLS) directed towards the revision of a substantial segment of the Classification proof.There are two principal strategies at present directed towards a Classification proof.The one employed in the first successful proof and also, with certain modifications, in the GLS proof, I shall refer to as the Semisimple Approach to the Classification.The other, which has been the object of considerable activity recently, I shall refer to as the Unipotent Approach to the Classification.Each has its advantages and its drawbacks and neither is, at present, completely independent of the other.In unison they provide a complete proof of the Classification Theorem.A question at present is the natural domain for each of these methods.Of course the future may bring entirely new and wonderful approaches to the subject.The modern history of the Classification began around 1950 when several mathematicians-notably Brauer, Suzuki and Wall-began to investigate simple groups of even order satisfying certain local conditions.This work eventually congealed into the Brauer-Suzuki-Wall Theorem [BSW] characterizing the two-dimensional projective special linear groups over finite fields.Brauer in particular championed the strategy of characterizing finite simple groups of even order by the centralizer of an involution.Suzuki, on the other hand, established the nonexistence of finite simple CA-groups of odd order [S1].(A group G is a CA-group if the centralizer of every nonidentity element of G is abelian.)This result was the inspiration for the Feit-Thompson Theorem proving the nonexistence of nonabelian finite simple groups of odd order.Meanwhile Suzuki pursued the classification of transitive permutation groups of odd degree in which the stabilizer of a point has a regular
We give a characterization of the finite groups having nilpotent or abelian Hall $\pi$-subgroups that can easily be verified using the character table.
We give a characterization of the finite groups having nilpotent or abelian Hall $\pi$-subgroups that can easily be verified using the character table.
Abstract. The non-commuting graph of a finite group G is a highly symmetrical object (indeed, embeds in ), yet its complexity pales in comparison to that of G . Still, โฆ
Abstract. The non-commuting graph of a finite group G is a highly symmetrical object (indeed, embeds in ), yet its complexity pales in comparison to that of G . Still, it is natural to seek conditions under which G can be reconstructed from . Surely some conditions are necessary, as is evidenced by the minuscule example . A conjecture made in [J. Algebra 298 (2006), 468โ492], commonly referred to as the AAM Conjecture, proposes that the property of being a nonabelian simple group is sufficient. In [Sib. Math. J. 49 (2008), no. 6, 1138โ1146], this conjecture is verified for all sporadic simple groups, while in [J. Algebra 357 (2012), 203โ207], it is verified for the alternating groups. In this paper we verify it for the simple groups of Lie type, thereby completing the proof of the conjecture.
In 1983, Danny Gorenstein announced the completion of the Classification of the Finite Simple Groups.This announcement was somewhat premature.The Classification of the Finite Simple Groups was at last completed with โฆ
In 1983, Danny Gorenstein announced the completion of the Classification of the Finite Simple Groups.This announcement was somewhat premature.The Classification of the Finite Simple Groups was at last completed with the publication in 2004 of the two monographs under review here.These volumes, classifying the quasithin finite simple groups of even characteristic, are a major milestone in the history of finite group theory.It is appropriate that the great classification endeavor, whose beginning may reasonably be dated to the publication of the monumental Odd Order Paper [FT] of Feit and Thompson in 1963, ends with the publication of a work whose size dwarfs even that massive work.
In this paper, Conway's group, Co3, is characterized among simple groups of even type with e(G) = 3, by a restriction on the p-local structure for some odd prime p โฆ
In this paper, Conway's group, Co3, is characterized among simple groups of even type with e(G) = 3, by a restriction on the p-local structure for some odd prime p for which m2,p(G) = 3. 2000 Mathematics Subject Classification 20D05.
In 1960, Paul A. Smith asked the following question.If a finite group G acts smoothly on a sphere with exactly two fixed points, is it true that the tangent G-modules โฆ
In 1960, Paul A. Smith asked the following question.If a finite group G acts smoothly on a sphere with exactly two fixed points, is it true that the tangent G-modules at the two points are always isomorphic?We focus on the case G is an Oliver group and we present a classification of finite Oliver groups G with Laitinen number a G = 0 or 1.Then we show that the Smith Isomorphism Question has a negative answer and a G โฅ 2 for any finite Oliver group G of odd order, and for any finite Oliver group G with a cyclic quotient of order pq for two distinct odd primes p and q .We also show that with just one unknown case, this question has a negative answer for any finite nonsolvable gap group G with a G โฅ 2.Moreover, we deduce that for a finite nonabelian simple group G, the answer to the Smith Isomorphism Question is affirmative if and only if a G = 0 or 1.
We present some highlights of the 110-year project to classify the finite simple groups.
We present some highlights of the 110-year project to classify the finite simple groups.
In recent years great progress has been made toward the classification of finite simple groups in terms of local subgroups and in particular the centralizers of involutions.If this program is โฆ
In recent years great progress has been made toward the classification of finite simple groups in terms of local subgroups and in particular the centralizers of involutions.If this program is to be completed one must show that an arbitrary simple group G possesses an involution for which Ca(t) is isomorphic to a centralizer in a known simple group.This paper concerns itself with that problem for simple groups of component type; that is groups G such that E(C(t)/O(C(t)))for some involution in G.These include most of the Chevalley groups of odd characteristic, most of the alternating groups, and many of the sporadic simple groups.D. Gorenstein has conjectured that in a group of component type, the centralizer of some involu- tion is usually in a "standard form."A proof is supplied here of a portion of that conjecture.To be more precise, define a subgroup K of a finite group G to be tightly embedded in G if K has even order while K c K g has odd order for each g G-N(K).Define a quasisimple subgroup .4 of G to be standard in G if [.4,.4g] for each g G, K Ca(,4) is tightly embedded in G, and N(A) N(K).Let G be a finite simple group of component type in which O,,(C(t)) O(C(t))E(C(t)) for each involution in G. Let A be a "large component."Then it is shown, modulo a certain special case where A has 2-rank l, that A is standard in G in the sense defined above.Other theorems establish properties of tightly embedded subgroups.They show that, under the hypothesis of the last paragraph, the centralizer of each involution centralizing A contains at most one component distinct from A, and that component must have 2-rank if it exists.Further, it can be shown that the 2-rank of the centralizer of A is bounded by a function of A, which seems to be or 2 if A is not of even characteristic.Proofs of the various theorems utilize properties of the Generalized Fitting Subgroup F*(G) of a group G, developed by Gorenstein and Walter.These properties appear in Section 2. Also important to the proof is the classification of groups with dihedral Sylow 2-groups, Alperin's fusion theorem, the recent result on 2-fusion due to Goldschmidt, and Theorem 3.3 in Section 3, which extends Bender's classification of groups with a strongly embedded subgroup.Statements of the major theorems appear in Section l, along with a brief explanation of notation.
THEOREM I. Let G be a finite group with F*(G) simple. Let z be an involution in G and K a subnormal subgroup of CQ(z) such that K has nonabelian โฆ
THEOREM I. Let G be a finite group with F*(G) simple. Let z be an involution in G and K a subnormal subgroup of CQ(z) such that K has nonabelian Sylow 2-subgroups and z is the unique involution in K. Assume for each 2-element k C K that kG n C(z) C N(K) and for each g e C(z) N(K), that [K, Kg]<! O(C(z)). Then F*(G) is a Chevalley group of odd characteristic, M11, M12 or SP6(2). COROLLARY II. Let G be a finite group with F*(G) simple and let K be tightly embedded in G such that K has quaternion Sylow 2-subgroups. Then F*(G) is a Chevalley group of odd characteristic, M11, or M12. COROLLARY III. Let G be a finite group with F*(G) simple. Let z be an involution in G and K a 2-component or solvable 2-component of CQ(z) of 2-rank 1, containing z. Then F*(G) is a Chevalley group of odd characteristic or M,1. Theorem I follows from Theorems 1 through 8, stated in Section 2, which supply more specific information under more general hypotheses. Corollary II follows directly from Theorem I. Corollary III follows from Theorem I and Theorem 3 in [3]. All Chevalley groups with the exception of L2(q) and 2G2(q) satisfy the hypotheses of Theorem I. Terminology and notation are defined in Section 2. The possibility of such a theorem was first suggested by J.G. Thompson in January 1974, during his lectures at the winter meeting of the American Mathematical Society in San Francisco. At the same time Thompson also pointed out the significance of a certain section of the group, which is crucial to the proof. We have taken the liberty of referring to this section as the Thompson group of G; see Section 2 for its definition. The theorem finds its motivation in the study of component-type groups. Some applications to this theory are described in [6]. The remainder of this
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 โฆ
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.
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 โฆ
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.
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 โฆ
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).
H. Benderโs 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 โฆ
H. Benderโs 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.
This paper presents results on Schur multipliers of finite groups of Lie type. Specifically, let <italic>p</italic> denote the characteristic of the finite field over which such a group is defined. โฆ
This paper presents results on Schur multipliers of finite groups of Lie type. Specifically, let <italic>p</italic> denote the characteristic of the finite field over which such a group is defined. We determine the <italic>p</italic>-part of the multiplier of the Chevalley groups <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G 2 left-parenthesis 4 right-parenthesis comma upper G 2 left-parenthesis 3 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <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>G</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>3</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{G_2}(4),{G_2}(3)</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 F 4 left-parenthesis 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">{F_4}(2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the Steinberg variations; the Ree groups of type <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 4"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">{F_4}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and the Tits simple group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="squared upper F 4 left-parenthesis 2 right-parenthesis prime"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi /> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>4</mml:mn> </mml:msub> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mo>โฒ</mml:mo> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">^2{F_4}(2)โ</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.
Abstract : This study considers a class of doubly transitive groups satisfying the condition that the identity is the only element leaving three distinct letters fixed. The main object of โฆ
Abstract : This study considers a class of doubly transitive groups satisfying the condition that the identity is the only element leaving three distinct letters fixed. The main object of the investigation is to classify the groups which do not contain a regular normal subgroup of order 1 + N in case N is even. (Author)
Let <italic>G</italic> be a finite group with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis upper G right-parenthesis equals 1"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> โฆ
Let <italic>G</italic> be a finite group with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis upper G right-parenthesis equals 1"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>G</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">O(G) = 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <italic>A</italic> a standard component of <italic>G</italic> and <italic>X</italic> the normal closure of <italic>A</italic> in <italic>G</italic>. Furthermore, assume that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C left-parenthesis upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has cyclic Sylow 2-subgroups. Then conditions are given on <italic>A</italic> which imply that either <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper X equals upper A"> <mml:semantics> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">X = A</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 C left-parenthesis upper A right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>C</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>A</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">C(A)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has Sylow 2-subgroups of order 2. These results are then applied to the cases where <italic>A</italic> is isomorphic to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="dot 0"> <mml:semantics> <mml:mrow> <mml:mo>โ </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\cdot 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or <italic>Ru</italic>, proper 2-fold covering of the Rudvalis group.
All groups considered in this paper are finite.In current standard terminology, a group L such that L E and L/O(L) is quasisimple is said to be 2-quasisimple.Also any subnormal 2-quasisimple โฆ
All groups considered in this paper are finite.In current standard terminology, a group L such that L E and L/O(L) is quasisimple is said to be 2-quasisimple.Also any subnormal 2-quasisimple subgroup of a group G is called a 2-component of G.Recently, a great deal of progress has been made on the fundamental problem of classifying all finite groups G such that O(G)and such that G contains an involution such that H Co(t and[13]).These results suggest the importance of investigating such groups G in which Cu(L/O(L)) has 2-rank 1.Of particular interest is the case where L is of dihedral type.In [9], R. Solomon and the author obtained some results on groups G with O(G)and containing an involution G Z(G) such that H Ca(t) contains a 2-component L such that a Sylow 2-subgroup of L is dihedral, m2(CH(L/O(L)))-and such that Nu(L)/(LCu(L/O(L))) is cyclic.In this paper, the methods of [9] are applied to the case in which NH(L)/(LCu(L/O(L)))is not cyclic.The first main result of this paper is the following.THEOREM 1.Let G be a finite group with O(G) 1. Suppose the involution G-Z(G) is such that H Ca(t) contains a 2-component L such that a Sylow 2-subgroup oJ'L is dihedral, m2(Cu(L/O(L)))= and such that Nn(L)/(LCu(L/O(L))) is not cyclic.Let S Syl2(Na(L)) be such that S and let D S L. Then the following conditions hold:(i) L/O(L) is isomorphic to PSL(2, q2)for some oddprimepower q, Na(L) O(Na(L))H and S Sy/2(H).(ii) O2(G)= F(G)= Ca(E(G))= land F*(G)-E(G).(iii) If F*(G) is not simple, then F*(G) R x R where R is simple and L <rrtlrR> R.(iv) If F*(G) is simple and r2(F*(G)) < 4, then the possibilities for F*(G) and G can be obtained from [6, Main Theorem].
By W. Carter Roger: pp. viii, 331. ยฃ7.50. John Wiley & Sons, December 1972.)
By W. Carter Roger: pp. viii, 331. ยฃ7.50. John Wiley & Sons, December 1972.)
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,(ยฎ) 395 5.3.Groups โฆ
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,(ยฎ) 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]
In this paper we supply one link in a chain of results which will prove the following two conjectures: then [L(H),O(H)]QO(G). Unbalanced Group Conjecture. If G is a finite group โฆ
In this paper we supply one link in a chain of results which will prove the following two conjectures: then [L(H),O(H)]QO(G). Unbalanced Group Conjecture. If G is a finite group with O(C ff (ฮฏ)) g O(G) for some involution teG, then O(C G (t)) acts nontrivially on L/Z*(L) where L is a 2-component of G with L/Z*(L) isomorphic to one of the following simple groups:(1) A simple Chevalley group or twisted variation over a field of odd order;(2) An alternating group of odd degree;(3) PSL (3,4) of He, the simple group of Held.
In [7] and [8], Tutte considered a vertex-transitive group of automorphisms of a finite, connected, trivalent graph. He showed that if the stabilizer of a is on adjacent vertices, then โฆ
In [7] and [8], Tutte considered a vertex-transitive group of automorphisms of a finite, connected, trivalent graph. He showed that if the stabilizer of a is on adjacent vertices, then its order divides 3 . 24. As observed by Sims [5], the hypothesis is equivalent to the following group-theoretic conditions: a) G is a finite group generated by a pair of subgroups {P1, Pd, b) i Pi: Pi n P2 1= 3 for i = 1, 2, c) no non-trivial normal subgroup of G is contained in P1 A, d) P1 and P2 are G-conjugate. What happens if we drop condition d) or, what is essentially the same thing, replace vertex transitive by edge transitive? This question is primarily motivated by the examples afforded by the rank 2 BN pairs over GF(2). In this case, the trivalent graph mentioned above is the so-called building associated to the BN pair [6]. In this paper, we classify all pairs of subgroups (P1, P2) for which hypotheses a), b) and c) are satisfied. There are precisely fifteen such pairs, and in particular, we find that P1 n P2 has order dividing 27. In order to describe the results more completely, let us define an amalgam to be a pair of group monomorphisms (5, 02) with the same domain: P, 1 B P2. We will say that (01, 02) is finite if both co-domains P1, P2 are finite. In this case, we define the index of the amalgam to be the pair of indices (I Pl: im s1 I, P2: im 02 1). By a completion of the amalgam we mean a pair of homomorphisms (*1, '2) to some group G making the obvious diagram commute, i.e., such that 01* = 102'2 (in right-hand notation). By abuse of notation, we may say that G is a completion of the amalgam. Of course, we always have the trivial completion g1 = g2 = 1We also always have the universal completion, usually known as the amalgamated product, from