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Walter Feit (October 26, 1930 – July 29, 2004) was an Austrian-American mathematician renowned for his groundbreaking work in finite group theory. One of his most famous results, achieved in collaboration with John G. Thompson, is the Feit–Thompson theorem (also known as the Odd Order Theorem), which established that every finite group of odd order is solvable. He spent a significant portion of his career as a professor at Yale University, where he influenced and mentored many students in algebra and group theory.

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Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals a squared plus left-parenthesis 2 b right-parenthesis squared"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals a squared plus left-parenthesis 2 b right-parenthesis squared"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>a</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mi>b</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">p = a^{2} + (2b)^{2}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be a prime. It is shown that each of the two Diophantine equations <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x squared minus p y squared equals a"> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>−</mml:mo> <mml:mi>p</mml:mi> <mml:msup> <mml:mi>y</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>a</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x^{2}-py^{2} =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="4 b"> <mml:semantics> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>b</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">4b</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has integral solutions.
Let $G$ be a finite group with a faithful rational valued character of degree $n$. A theorem of I. Schur gives a bound for the order of $G$ in terms … Let $G$ be a finite group with a faithful rational valued character of degree $n$. A theorem of I. Schur gives a bound for the order of $G$ in terms of $n$, generalizing an earlier result of H. Minkowski who showed that the same bound holds if $G\subseteq GL(n,\mathbf {Q})$. This note contains strengthened versions of these results which in particular show that a $2$-subgroup of $GL(n,\mathbf {Q})$ of maximum possible order contains a reflection.
Let n be an even integer not divisible by 3 .Suppose that p = n2+n+l is a prime and 2"+1 = 1 (mod p) .The question is asked whether this … Let n be an even integer not divisible by 3 .Suppose that p = n2+n+l is a prime and 2"+1 = 1 (mod p) .The question is asked whether this can only occur if « is a power of 2 .It is noted that an affirmative answer to this question implies that a finite projective plane with a flag transitive collineation group is Desarguesian.
Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an even integer not divisible by 3. Suppose that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals n … Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be an even integer not divisible by 3. Suppose that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p equals n squared plus n plus 1"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>n</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p = {n^2} + n + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a prime and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 Superscript n plus 1 Baseline identical-to 1 left-parenthesis mod p right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mn>2</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo>≡<!-- ≡ --></mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo lspace="thickmathspace" rspace="thickmathspace">mod</mml:mo> <mml:mi>p</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{2^{n + 1}} \equiv 1\left ( {\bmod p} \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The question is asked whether this can only occur if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a power of 2. It is noted that an affirmative answer to this question implies that a finite projective plane with a flag transitive collineation group is Desarguesian.
If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</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="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are integers greater than … If <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a"> <mml:semantics> <mml:mi>a</mml:mi> <mml:annotation encoding="application/x-tex">a</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="m"> <mml:semantics> <mml:mi>m</mml:mi> <mml:annotation encoding="application/x-tex">m</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are integers greater than 1, then a large Zsigmondy prime is a prime <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l"> <mml:semantics> <mml:mi>l</mml:mi> <mml:annotation encoding="application/x-tex">l</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l vertical-bar a Superscript m Baseline minus 1 comma l does-not-divide a Superscript i Baseline minus 1"> <mml:semantics> <mml:mrow> <mml:mi>l</mml:mi> <mml:mrow> <mml:mo>|</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mi>l</mml:mi> <mml:mo>∤<!-- ∤ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>a</mml:mi> <mml:mi>i</mml:mi> </mml:msup> </mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mo fence="true" stretchy="true" symmetric="true" /> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">l\left | {{a^m} - 1,l\nmid {a^i} - 1} \right .</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="1 less-than-or-slanted-equals i less-than-or-slanted-equals m minus 1"> <mml:semantics> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>⩽<!-- ⩽ --></mml:mo> <mml:mi>i</mml:mi> <mml:mo>⩽<!-- ⩽ --></mml:mo> <mml:mi>m</mml:mi> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">1 \leqslant i \leqslant m - 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and either <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="l squared vertical-bar a Superscript m Baseline minus 1"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>l</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> </mml:msup> </mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mo fence="true" stretchy="true" symmetric="true" /> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{l^2}\left | {{a^m} - 1} \right .</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="l greater-than m plus 1"> <mml:semantics> <mml:mrow> <mml:mi>l</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mi>m</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">l &gt; m + 1</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The main result of this paper lists all the pairs <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis a comma m right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mi>m</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left ( {a,m} \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for which no large Zsigmondy prime exists.
If $a$ and $m$ are integers greater than 1, then a large Zsigmondy prime is a prime $l$ such that $l\left | {{a^m} - 1,l\nmid {a^i} - 1} \right .$ … If $a$ and $m$ are integers greater than 1, then a large Zsigmondy prime is a prime $l$ such that $l\left | {{a^m} - 1,l\nmid {a^i} - 1} \right .$ for $1 \leqslant i \leqslant m - 1$ and either ${l^2}\left | {{a^m} - 1} \right .$ or $l > m + 1$. The main result of this paper lists all the pairs $\left ( {a,m} \right )$ for which no large Zsigmondy prime exists.
(1986). Finite Linear Groups, the Commodore 64, Euler and Sylvester. The American Mathematical Monthly: Vol. 93, No. 9, pp. 717-719. (1986). Finite Linear Groups, the Commodore 64, Euler and Sylvester. The American Mathematical Monthly: Vol. 93, No. 9, pp. 717-719.
in Boston, Massachusetts at the age of 76.His death, after a short illness, was unexpected and came as a shock to his friends.During the past decade he had had several … in Boston, Massachusetts at the age of 76.His death, after a short illness, was unexpected and came as a shock to his friends.During the past decade he had had several serious ailments but he had always recovered satisfactorily.There was no reason to believe that he would not recover from this final illness.He was mentally alert and mathematically active to the end, and this has made it all the more difficult to realize that he is now gone.Brauer was born in Berlin on February 10, 1901.He was the youngest of three children of Max and Lilly Caroline Brauer.He showed an early interest in science and mathematics which was stimulated by his brother Alfred, who was older by seven years.He graduated from high school in September, 1918.After graduation he and his classmates were drafted for civilian service in Berlin, As the first World War ended two months later, his war time service was brief and did not seriously interrupt his education.In contrast, his older brother Alfred spent four years in the German Army during the war and was seriously wounded.In February 1919 he began his college education at the Technical University of Berlin.He soon realized that his interests were more theoretical than practical, and after one term he transferred to the University of Berlin.With the exception of a term at the University of Freiburg, he stayed at the University of Berlin until he was awarded his Ph.D. summa cum laude on March 16,1926.It was a custom at that time for students at one German University to spend some time at another.Brauer decided to spend a term at Freiburg.During that term he took a course on invariant theory
Let G be a finite simple group and let F be an algebraically closed field. A faithful projective F -representation of G of smallest possible degree often cannot be lifted … Let G be a finite simple group and let F be an algebraically closed field. A faithful projective F -representation of G of smallest possible degree often cannot be lifted to an ordinary representation of G, though it can of course be lifted to an ordinary representation of some central extension of G. It is a natural question to ask whether by considering non-central extensions, it is possible in some cases to decrease the smallest degree of a faithful projective representation.

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CHAPTER II Preliminary Lemmas of Lie TypeHypothesis 6.1.(i) p is a prime, ^ is a normal S p -subgroup of tyU, and U is a non identity cyclic p'-group.(ii) C … CHAPTER II Preliminary Lemmas of Lie TypeHypothesis 6.1.(i) p is a prime, ^ is a normal S p -subgroup of tyU, and U is a non identity cyclic p'-group.(ii) C U CP) = 1.(iii) $P' is elementary abelian and ^P' S Z($).(iv) \W\ is odd.Let U = <tf>, |U| = u, and |5(S: D(SP)| = p\ Let ^ be the Lie ring associated to *P ([12] p. 328).Then Sf= -Sf*© J2f where JSf* and -Sf correspond to W and 5p' respectively.Let jgf = Jg?7p_2f *.For i = 1,2, let 17* be the linear transformation induced by U on -£?.LEMMA 6.1.Assume that Hypothesis 6.1 is satisfied.Let s lf • • •, s n 6e £/ &e characteristic roots of U lm Then the characteristic roots of U 2 are found among the elements £<£, -with 1 ^ i < j ^ n.Proof.Suppose the field is extended so as to include e lf • • •, e n .Since U is a p'-group, it is possible to find a basis x u • • •, x n of -Sf such that XiU x = e t x i9 X ^i gw.Therefore, xJI^XiUx = 6^0^-0?/.As U induces an automorphism of j£f f this yields that Since the vectors XrXj with i < j span .Sf, the lemma follows.By using a method which differs from that used below, M. Hall proved a variant of Lemma 6.2.We are indebted to him for showing us his proof.LEMMA 6.2.Assume that Hypothesis 6.1 is satisfied, and that Ui acts irreducibly on -Sf.Assume further that n = q is an odd prime and that U x and U 2 have the same characteristic polynomial.Then q > 3 and u < 3« /aProof.Let e pi be the characteristic roots of U l9 0 ^ i < n.By Lemma 6.1 there exist integers i, j, k such that e»*e» y = £**.Raising this equation to a suitable power yields the existence of integers a and b with 0 ^ a < b < q such that s" a+p6 -1 = 1.By Hypothesis 6.1 (ii), the preceding equality implies p a + p b -1 = 0(mod u).Since U X acts irreducibly, we also have p q -1 = 0(mod u).Since U is a p'-group, 789 790 SOLVABILITY OF GROUPS OF ODD ORDER ab =£ 0. Consequently, P a + P b -l = 0(mod u), p 9 -1 = 0(mod u), 0 < a < b < q .Let d be the resultant of the polynomials / = x a + x b -1 and gx q -1.Since q is a prime, the two polynomials are relatively prime, so d is a nonzero integer.Also, by a basic property of resultants, (6.2) d = hf+kg for suitable integral polynomials h and k.Let e g be a primitive gth root of unity over «^, so that we alsohave ej' + ej* -1) jGt (sp fl + sp 6 -1) ; = II {3 + e; (a " 6) + For g = 3, this yields that d 3 = (3 -1 + l+l) 2 = 4 2 , so that d = ±4.Since it is odd (6.1) and (6.2) imply that u = l.This is not the case, so q > 3.Each term on the right hand side of (6.3) is non negative.As. the geometric mean of non negative numbers is at most the arithmetic mean, (6.3) implies thatThe algebraic trace of a primitive gth root of unity is -1, hence d 21 ' ^ 3 .Now (6.1) and (6.2) imply that u^ \d\ ^&12 .Since 3 9/2 is irrational, equality cannot hold.LEMMA 6.3.If ^ is a p-group and W = D($), then C n (^)/C H is elementary abelian for all n.Proof.The assertion follows from the congruence [A u • • -, A n ] p = [A u • •., A n .u A p n ](mod C. +l (SP)) , valid for all A l9 • • •, A n in ^P.LEMMA 6.4.Suppose that a is a fixed point free p'-automorphism-
LEMMA 11.6.Suppose that Hypothesis 11.3 is satisfied.Assume further that a is odd and p = 3.Then Sf is coherent. LEMMA 11.6.Suppose that Hypothesis 11.3 is satisfied.Assume further that a is odd and p = 3.Then Sf is coherent.
of Results Proved in Chapter IVIn this chapter, we begin the proof of the main theorem of this paper.The proof is by contradiction.If the theorem is false, a minimal counterexample … of Results Proved in Chapter IVIn this chapter, we begin the proof of the main theorem of this paper.The proof is by contradiction.If the theorem is false, a minimal counterexample is seen to be a non cyclic simple group all of whose proper subgroups are solvable.Such a group is called a minimal simple group.Throughout the remainder of this chapter, © is a minimal simple group of odd order.We will eventually derive a contradiction from the assumed existence of ©.In this section, the results to be proved in this chapter are summarized.Several definitions are required.Let 7T* be the subset of n(®) consisting of all primes p such that if sp is a Sp-subgroup of ®, then either Sfif^ift) is empty or $P contains a subgroup 21 of order p such that C^(2l) = 21 x 95 where 33 is cyclic.Let n? be the subset of TC* consisting of those p such that if ^5 is a Sp-subgroup of © and a is the order of a cyclic subgroup of iV($P)/$pC($P), then one of the following possibilities occurs:(i) a divides p -1. (ii) $P is abelian and a divides p + 1. (iii) | $P | = p z and a divides p + 1.We now define five types of subgroups of ©.The basic property shared by these five types is that they are all maximal subgroups of ©.Thus, for x = 1, II, III, IV, V, any group of type x is by definition a maximal subgroup of @.The remaining properties are more detailed.We say that 9K is of type I provided (i) 3K is of Frobenius type with Frobenius kernel §. (ii) One of the following conditions is satisfied: (a) § is a T. I. set in ©.(b) 7r(£)S7r*.(c) & is abelian and m( §) = 2. (iii) If p e 7r(3K/ §), then m p (2TC) ^ 2 and a S P -subgroup of 2Ji is abelian.The remaining four types are by definition three step groups.If @ is a three step group, we use the following notation: @ = ©'2^ , ©' n 25*! = 1 , C&iZ&J = SB*.*•• • • Furthermore, !Q denotes the maximal normal niljxrtent S-subgroup of ©.By definition, §^S' so WG let II be a complement for § in ©', 845 846 SOLVABILITY OF GROUPS OF ODD ORDERIn addition to being a three step group, each of the remaining four types has the property that if 2Q 0 is any non empty subset of aO^aOS, -SB 1 -2B 2 , then iV @ (2B 0 ) = SJyBB,, by definition.The remaining properties are more detailed.We say that @ is of type II provided (i) U =£ 1 and U is abelian.(ii) N 9 (O)&&.(iii) iV®(2I) g @ for every non empty subset SI of @" such that (iv) | 2&! | is a prime.(v) For every prime p, if 2t 0 , 2tx are cyclic p-subgroups of U which are conjugate in © but are not conjugate in @, then either C^(2I 0 ) = 1 or C c (8y = 1.(vi) £>C(£>) is a T. I. set in @.We say that @ is of type III provided (ii) in the preceding definition is replaced by (ii)' Ai(U)S© f and the remaining conditions hold.We say that @ is of type IV provided (i) and (ii) in the definition of type II are replaced by (i)" U'*l, (ii)" AT<,(U)S© f and the remaining conditions hold.We say that <S is of type V provided (i) U = l.(ii) One of the following statements is true:(a) @' is a T. I. set in ©.(b) @' = sp x @ Of where @ 0 is cyclic and ty is a S p -subgroup of © with THEOREM 14.1.Let © 6e a minimal simple group of odd order. Two elements of a nilpotent S-subgroup § of © are conjugate in © i/ and only if they are conjugate in iV(£>). Either (i) or (ii) is true:(i) Every maximal subgroup of © is of type I. (ii) (a) © contains a cyclic subgroup 23 = SBx x 2B 2 wi</& £fte property that JV(2B 0 )=2B /or ever]/ non empty sw&set 2B 0 o/ SB-2$^-SB!.AZso, SB, * 1, i = 1, 2. (b) © contains maximal subgroups @ and 2 not o/ type 7 e 14. STATEMENT OF RESULTS PROVED IN CHAPTER IV 847 (c) Every maximal subgroup of ® is either conjugate to @ or X or is of type I.(d) Either @ or Z is of type II.(e) Both @ and % are of type II, III, IV, or V. (They are not necessarily of the same type.)In order to state the next theorem we need further notation.If 8 is of type I, let 8 = 2, = U C 2 (H) , where £> is the Frobenius kernel of 8.If 8 is of type II, III, IV, or V, we write 8 = 8'SB^ 8' n SBi = 1.Let £> be the maximal normal nilpotent S-subgroup of 8, let 11 be a complement for § in 8' and set SB = C^SBx), 2B a = 2B n 8', 3& = 2B -If 8 is of type II, let 8 = U C 2 ,(H) .ne §* If 8 is of type III, IV, or V, let 8 = 8'.If 8 is of type II, III, IV, or V, let 8 t = 8 U U Lzest We next define a set s/ = J^(8) of subgroups associated to 8. Namely, 3Ji e sf if and only if 2JI is a maximal subgroup of © and there is an element L in 8* such that C(L) g; 8 and C(L) g; 9Ji.Let {??!, • • •, S JJJ be a subset of s*/ which is maximal with the property that 9^ and Sfty are not conjugate if i =£ i.For 1 ^ i g w, let £>* be the maximal normal nilpotent S-subgroup of %.THEOREM 14.2.If 8 is of type 7, //, ///, IV, or V, then 8 and 8 X are tamely imbedded subsets of © with JV(S) = N(k) = 8 .JTf J^(8) is empty, 8 and 8 X are T. I. sets in ©. // j^(8) is non empty, the subgroups $ lf • • •, & n are a system of supporting subgroups for 8 and for 8T he purpose of Chapter IV is to provide proofs for these two theorems.Proof.The proof of (i) proceeds by a series of reductions.If 21 = 1, the theorem is vacuously true, so we may assume 21 =£ 1. Choose Z in Z(^5), and let G* be any element of M(2l; q) which is centralized by Z.By Lemmas 7.4 and 7.8, if S is any proper subgroup of £ containing 21Q*, then £>* C O P ,(2).Now let D* denote any element of M(2t; g) and let 8 be a proper subgroup of X containing 2IQ*.We will show that D* S O P <(8).First, suppose Z(?P) is non cyclic.Then Q* = <C D *(Z) |ZeZ(^)*>, so by the preceding paragraph, £>* g O P >(8).We can suppose that Z(?$) i& cyclic.Let Z be an element of Z(^P) of order p.We only need to show that [O*, Z] S O P '(S), by the preceding paragraph.Replacing £>* by [O*, Z], we may suppose that £>* = [£>*, Z].Furthermore, we may suppose that 21 acts irreducibly on D*/Z)(Q*).Suppose ZeO P ,, p (2).Then JQ* = [O*, Z] s 0,^(8) nD*S O P ,(S> and we are done.If 21 is cyclic, then Z is necessarily in O P ', P (8), 17.A DOMINATION THEOREM AND SOME CONSEQUENCES 851 since 21 fl O p > >p (8) =£ 1.Thus, we can suppose that 21 is non cyclic.Let 21, = C a (£l*) = C a (Q*/Z>(O*)) f so that 21/21! is cyclic and Zg2I lB We now choose W of order p in 2I X such that <Z, W> <$p.Suppose by way of contradiction that O* g£O P <(8).Then by Lemma 7.8, we can find a subgroup $ of 2IC(2I 1 ) which contains 2I£>* and such that £>*gO P .(ft).In particular, D* g O P ,(C(TF)).Thus, we suppose without loss of generality that 2 = C(Tr).Let ^P* be a S psubgroup of 8 which contains *p = *p n C(TF).If ty* = $, then ZeO p >,p(2), by Lemma 1.2.3 of [21], which is not the case.Hence, $ is of index p in ^P*.Clearly, 21 £ $ and ZeZ$>).Hence, HP*, ^] £ Z($) £ 21.Let SJJf = *P* n 0 P <, P (S) so that W is a S p -subgroup of O P ,, P (8).Then RJf, <£>, Q*] £ [21, £>*] n O P ,.P (8) £ Q* n O P ,.P (S), so that [5ft*, <Z>, O*] £ 0^(8).Let 93 = O P ,.P (S)/O P ,(S) and let ^ = C^O*).The preceding containment implies that [53, (Zy\ g 53 lB Let S3 2 = JVSB^I)-Then ^ acts trivially on the Q*2l-admissible group SB^.Hence, so does [<Z>, O*] = D*, that is, S3 2 £ SB l B This implies that S3 = 5$! is centralized by Q* so Q* g O P (S).We have succeeded in showing that if Q* is in M(2I; q) and 8 is any proper subgroup of X containing WO.*, then D* £ 0,,(S).Now let ^, • • •, <^ be the orbits under conjugation by C(2l) of the maximal elements of M(2I; q).We next show that if De <^, Dx G ^ and i =£ i, then JD n &i = 1.Suppose false and i, j, O f C^ are chosen so that | Q n Oi I is maximal.Let £1* = iV Q (£} n Ox) and Of = NQJJOL fl C^).Since D and C^ are distinct maximal elements of M(2I; g), QnDiisa proper subgroup of both Q* and Of.Let 2 = JV(D n Oi).By the previous argument, <X1*, Qi*> £ O P /(S).Let 5R be a S,-subgroup of O p >(2) containing d* and permutable with 21 and let 5Ri be a S 9subgroup of O p >(2) containing £i* and permutable with 21.The groups 31 and 3^ are available by D pq in 2IO P >(8).By the conjugacy of Sylow systems, there is an element C in O P '(8)2t such that W = 21 and SR° = 5RL AS 21 has a normal complement in O p /(8)2t, it follows that C centralizes 21.Let D be a maximal element of H(2I; q) containing Six.Then D n Qi 2 Of =) D fl Qi, and so De ^-.Also, Q n Q a 2 D*° D(Qfl O!) 0 so that D e ^ and i = j.To complete the proof of (i), let Q f C^ be maximal elements of M(2I; g) with Qe^^e ^.Suppose Ae21* and C C (A)^1, C Oi (A)^l.Let 8 = C(A), let 5R be a iS,-subgroup of O p >(2) containing C^(A) and permutable with 21, and let 9^ be a S 9 -subgroup of 0 P '(8) containing CfxJLA) and permutable with 21.Then W = % for suitable C in C(2I).Let D* be a maximal element of M(2I; g) containing 3^.Then D* n Oa 2 C Ol (A) gt 1 so Q* G ^-. Also, D* n D 0 2 (C D (A))* ^lso D* G tfj and i = j.This completes the proof of (i).As for (ii), if 21G ._5*if^($P),then there is an element A in 2t* 19.AN ^-THEOREM
The purpose of this chapter is to prove the following result.THEOREM 37.1.There are no groups ® which satisfy conditions (i)-(iv) of Theorem 27.1.Once it is proved, Theorem 37.1 together with … The purpose of this chapter is to prove the following result.THEOREM 37.1.There are no groups ® which satisfy conditions (i)-(iv) of Theorem 27.1.Once it is proved, Theorem 37.1 together with Theorem 27.1 will serve to complete the proof of the main theorem of this paper.In this chapter there is no reference to anything in Chapters II-V other than the statement of Theorem 27.1.The following notation is used throughout this chapter.© is a fixed group which satisfies conditions (i)-(iv) of Theorem 27.1.|U|u v -1 U* = C(U) and \VL*\=u*.U* = <C^>, U = Ur l% .Thus U = Do = [£}, «p*] so that 0 = Q*xQo.P and Q are fixed elements of sp** and £}** respectively.For any integer n > 0, ^n is the ring of integers mod n.If n is a prime power then ^~n is the field of n elements.U acts as a linear transformation on ^5.Let m(t) be the minimal polynomial of U on sp.Then m(t) is an irreducible polynomial of degree q over J^.Let co be a fixed root of m(t) in ^».Then a> is a primitive uth root of unity in ^pq and Q),o) p , •••,<w» flr ~1 are all the characteristic roots of U on $p.38.The Sets J^ and & LEMMA 38.1.There exists an element YefCfi such that $P* normalizes Proof.£}* normalizes U* and D* is contained in a cyclic subgroup of JV(U*) of order pq.Hence some element of order p in C(Q*) normalizes U*.Since C(Q*) = £$$* every subgroup of order p in C(D*) is of the form Y-^*Y for some Fe£V Hence it is possible to choose FeOo such that F" 1 ^* Y normalizes U*.Since [$*,U]E$, ion
Let G be a group which has a faithful representation as a transitive permutation group on m letters in which no permutation other than the identity leaves two letters unaltered, … Let G be a group which has a faithful representation as a transitive permutation group on m letters in which no permutation other than the identity leaves two letters unaltered, and there is at least one permutation leaving exactly one letter fixed. It is easily seen that if G has order mh, a necessary and sufficient condition for G to have such a representation is that G contains a subgroup H of order h which is its own normalizer in G and is disjoint from all its conjugates. Such a group G is called a Frobenius group of type ( h, m ).
In this paper we classify finite groups G with a faithful, quasiprimitive (see Notation), unimodular representation X with character χ of degree six over the complex number field. There are … In this paper we classify finite groups G with a faithful, quasiprimitive (see Notation), unimodular representation X with character χ of degree six over the complex number field. There are three gaps in the proof which are filled in by [16; 17]. These gaps concern existence and uniqueness of simple, projective, complex linear groups of order 604800, |LF(3, 4)|, and |PSL 4 (3)|. By [19] , X is a tensor product of a 2-dimensional and a 3-dimensional group, or a subgroup thereof, or X corresponds to a projective representation of a simple group, possibly extended by some automorphisms. The tensor product case is discussed in section 10. Otherwise, we assume that G/Z(G) is simple. We discuss which automorphisms of G/Z(G) extend the representation X (that is, lift to the central extension G and fix the character corresponding to X) just after we find X(G).
Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes … Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes et algebre de Lie, neuvieme Livre du traite, est consacre aux concepts fondamentaux pour les algebres de Lie. Il comprend les paragraphes: - 1 Definition des algebres de Lie; 2 Algebre enveloppante d une algebre de Lie; 3 Representations; 4 Algebres de Lie nilpotentes; 5 Algebres de Lie resolubles; 6 Algebres de Lie semi-simples; 7 Le theoreme d Ado. Ce volume est une reimpression de l edition de 1971.
Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the representations of … Our purpose here is to study the irreducible representations of semisimple algebraic groups of characteristic p 0, in particular the rational representations, and to determine all of the representations of corresponding finite simple groups. (Each algebraic group is assumed to be defined over a universal field which is algebraically closed and of infinite degree of transcendence over the prime field, and all of its representations are assumed to take place on vector spaces over this field.)
A finite transitive permutation group G is called a Frobenius group if every element of G other than 1 leaves at most one letter fixed, and some element of G … A finite transitive permutation group G is called a Frobenius group if every element of G other than 1 leaves at most one letter fixed, and some element of G other than 1 leaves some letter fixed. It is proved in [ 20 ] (and sketched below) that if k is a number field such that SL (2, 5) and one other nonsolvable group Ŝ 5 of order 240 are realizable as Galois groups over k, then every Frobenius group is realizable over k. It was also proved in [ 20 ] that there exists a quadratic (imaginary) field over which these two groups are realizable. In this paper we prove that they are realizable over the rationals Q , hence we Obtain THEOREM 1. Every Frobenius group is realizable as the Galois group of an extension of the rational numbers Q .
By focussing attention on indecomposable modular representations J. G. Thompson [11] has recently simplified and generalized some classical results of R. Brauer [1] concerning groups which have a Sylow group … By focussing attention on indecomposable modular representations J. G. Thompson [11] has recently simplified and generalized some classical results of R. Brauer [1] concerning groups which have a Sylow group of prime order. In this paper this approach will be used to prove some results which generalize theorems of R. Brauer [2] and H. F. Tuan [12].
The purpose of this paper is to determine the structure of some finite groups in which all Sylow subgroups of odd order are cyclic. This assumption on Sylow subgroups simplifies … The purpose of this paper is to determine the structure of some finite groups in which all Sylow subgroups of odd order are cyclic. This assumption on Sylow subgroups simplifies the structure of groups considerably, but the structure of 2-Sylow subgroups might be too complicated to make any definite statement on the structure of the groups. In this paper, therefore, we shall make another assumption on 2-Sylow subgroups, and our main result may be stated as follows. Let G be a non-solvable group of finite order. We assume that all Sylow subgroups of odd order are cyclic, and moreover that a 2-Sylow subgroup is either (a) a dihedral group, or (b) a generalized quaternion group. Then G contains a normal subgroup G1 such that [G: G1] ? 2 and G1 = Z X L, where Z is a solvable group whose Sylow subgroups are all cyclic, and L is isomor-phic with the linear fractional group LF (2, p) over the prime field of characteristic p in the case (a), and with the special linear group SL (2, p) in the case (b).
Let (5 be a group of finite order g. By a generalized character of (M, we shall mean a linear combination of the (ordinary) irreducible characters of (M with integral … Let (5 be a group of finite order g. By a generalized character of (M, we shall mean a linear combination of the (ordinary) irreducible characters of (M with integral rational coefficients. The main purpose of this paper is a characterization of the generalized characters. It will be convenient to call a group e an elementary group, if e is the direct product of a cyclic group I of order a and a p-group 3 of order pr where a is not divisible by the prime number p. We shall then prove THEOREM 1. A complex-valued function 0(G) defined on (5 is a generalized character of (5, if and only if the following two conditions are satisfied (I) 0 is a class function, i.e., the value of O(G) is constant for the elements of each class of conjugate elements of (X. (II) For every elementary subgroup e of (D, the restriction of 0 to e is a generalized character of A. The necessity of these conditions is obvious. The sufficiency is equivalent with a theorem on induced characters obtained in an earlier paper.' However, since the proof of this earlier theorem can be simplified considerably, I will prove Theorem 1 here without reference to the previous paper (Sections 2 and 3). As an immediate consequence of Theorem 1, we have THEOREM 2. The function 0 defined on @5 is an irreducible character of (M, if and only if besides conditions (I) and (II) of Theorem 1, the following further conditions are satisfied (III) The average value of I 0 12 is 1;
odd order are soluble. We shall use the term involution for a group element of order 2. If m is the total number of involutions of 65 and if we … odd order are soluble. We shall use the term involution for a group element of order 2. If m is the total number of involutions of 65 and if we set n = g/m, the same method shows that 65 contains a normal subgroup V distinct from 5 such that the index of V is either 2 or is less than [n(n + 2)/2]! (where [x] denotes the largest integer not exceeding the real number x). If J is an involution in 65 and if n(J) is the order of its normalizer 91(J) in 5, then n < n(J). It then follows that there exist only a finite number of simple groups in which the normalizer of an involution is isomorphic to a given group.
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).
This paper proves the following theorem: This paper proves the following theorem:
If there is given a subgroup 5 of a (finite) group G, we may ask what information is to be obtained about the structure of G from a knowledge of … If there is given a subgroup 5 of a (finite) group G, we may ask what information is to be obtained about the structure of G from a knowledge of the location of S in G. Thus, for example, famed theorems of Frobenius and Burnside give criteria for the existence of a normal subgroup N of G such that G = NS and 1 = N ⋂ S, and hence in particular for the non-simplicity of G. To aid in locating S in G, and to facilitate exploitation of the transfer, we single out a descending chain of normal subgroups of S. Namely, we introduce the focal series of S in G by means of the recursive formulae