CASTLE IN THE ATTIC BY ELIZABETH WINTHROP
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ON NORMAL-COMPOSITION SERIES WITH SIMPLE NONABELIAN FACTORS Tuval Foguel Department of Mathematics, The University of the West Indies, Mona, Kingston 7, Jamaica. E-mailaddress: foguel@uwimona.edu.jm
Abstract. In this paper we call a series of normal subgroups of a groupG a normal-composition series if
each subgroup in the series is normal inGthere is no refinement of this series with all subgroups, and
being normal inG. We show that a finite group that has a normal-composition series with simple nonabelian factors is a direct product of its simple factors. Notation
H≤G,H is a subgroup ofG.
Aut(G), Automorphism group ofGp. 130]. [4, Z(G), Center ofG[4, p.39]. Inn(G), Inner Automorphism group ofG(Inn(G)≅G/Z(G)) [4, p.130]. Out(G)= Aut(G)/Inn(G).C (A), Centralizer inGof the subgroupA[4, p.88]. G N (A), Normalizer inGof the subgroupA [4,p. 48]. G g-1H= gHgforH≤Gandg∈G. a H, ImageofH undera forH≤G anda∈Aut(G).Introduction Inthis paper we show that a finite group that has a normal-composition series with simple nonabelian factors is a direct product of its simple factors. This answers problem
1 † AMS1991Mathematics Subject Classification. Primary 20D99; Secondary 20E34.
5.57 of the Kourovka Notebook [2]. The proof is elementary once one uses Schreier Conjecture. In this paper all groups are assumed to be finite, and simple groups are assumed to be nonabelian. Definition.A series of subgroups of a groupG1 = G< G< .... < G= G 0 1n is acomposition seriesif there is no nontrivial refinement of this series. Definition.A series of normal subgroups of a groupGis called anormal-composition seriesif each subgroup in the series is normal inG, andthere is no nontrivial refinement of this series with all subgroups being normal inG. Theorem (Schreier Conjecture) [1, Theorem 1.46].If G isa nonabelian finite simple group, then Out(G)is solvable. Problem 5.57 of [2]is:a) Dose every extension of a finite nonabelian simple group by a finite nonabelian simple group split ? b) Must a semidirect product of two finite nonabelian simple groups decompose into a direct product ? Theorem 1 gives an affirmative answer to both problems.Theorems Theorem 1.Every extension of a finite nonabelian simple group by a finite nonabelian simple group decomposes into a direct product.
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Proof. Assume thatA is a normal nonabelian simple subgroup ofG and thatB≅G/A, andBa nonabelian simple group. is SinceA isa normal subgroup ofG,C (A) is normal inG. SinceA is a G nonabelian simple group,A∩C (A)≤ Z(A) = 1. Thus,/AAC (A)≅C (A) and G GG C (A)isomorphic to a normal subgroup of isB. SinceB is simpleB≅C (A), or G G C (A)= 1. AssumethatB≅C (A), thenG≅A x B. Assumethat= 1C (A), then G GG G =G/C (A) is isomorphic to a subgroup ofAut(A)[4, Theorem 7.1], thus since G Inn(A)≅A andB = G/A,B is isomorphicto a subgroup ofAut(A)/Inn(A )= Out(A). ButB is a nonabelian simple group andOut(A) is solvable by Schreier Conjecture, a contradiction. ■ Theorem 2.If G isfinite group that has a normal-composition series with simple nonabelian factors , thenG isa direct product of its simple factors. Proof. The groupG has a normal composition series 1 = G< G< .... < G< G= G,such thatA =G /Gis a nonabelian simple 0 1n n+1i ii-1 group, and eachG is normal inG. i
Proofby induction:n = 1is obvious. Let us assumeG≅.... x AA x. SinceA is n 1n 1
simpleC (A)∩A =1.SinceG is a direct product of theA’s,G <(A ),A Cand G 11 ni n1 G1 sinceA is a normal subgroup ofG,)C (Aa normal subgroup of isG. Thus 1 G1 G/A C(A ) is a factor ofG/Gisomorphic to a subgroup ofand isOut(A ), which is 1 G1 n,1 solvable, thusA C(A )= G. Andsince)C (A≅(A )/AA CC (A= G/A) isa 1 G1 G1 1G 11 1,G 1 group with a normal-composition series with simple nonabelian factors, )C (A≅A x.... x A, andG≅.... x AA x. G 12 n+1n+1 1■
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Definition.A subgroupH of a groupGis called acharacteristic subgroupofG iffor a alla∈Aut(G), H≤H. A groupG ischaracteristically simpleif it has no proper nontrivial characteristic subgroups. Definition.ARemak decompositionof a groupG is an expression ofG as a direct product of nontrivial subgroups with no refinement, i.e.G = Hx .... x Hand eachHis 1 niindecomposable as a direct product of nontrivial subgroups. Theorem 3.a finite group with a nonabelian characteristically simple normalIf G is subgroup A whichhas a composition series of length < 5, andG/A isa nonabelian simple group, thenG isisomorphic to a direct product of simple groups. Proof. Sincea nonabelian characteristically simple group,A isA=A x.... x Awhere 1 n n < 5and theAisomorphic nonabelian simple groups[3, Theorem 3.3.15].’s are i SinceZ(A) = 1,Ahas a unique Remak decomposition up to permutation of theA’s i g [3, Theorem 3.3.10]. Thus for allg∈Gand for anyAjA =i = 1 ,.., n,A ,, i i g g j =1 ,..., n, becauseA=.... x AA xis a Remak decomposition ofAsince. So 1 nA≤)N (A forG/Ai = 1 ,.., n ,the acts onApermuting them. So’s by G ii(G/A) =N (A)/A is a subgroup of index≤4 inG/A≅B. Thus,there exists a A Gi i homomorphism fromB to the Symmetric group on four elements [4, Theorem 3.18], which is solvable. SoN (A) = G and eachAis normal inG, thus by Theorem 2 G ii G≅A x.... x Ax B. 1 n■ Note 1. Theorem 3 is not true if the groupA has a composition series of length= 5, just let the alternating group of degree5permute5copies of the same finite nonabelian simple group.
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Note 2. Theorem 3 is not true if the groupA is abelian because then we do not have the uniqueness of the Remak decomposition. References 1) DanielGorenstein, FiniteSimple Groups: An Introduction to Their Classification (Plenum Press, New York and London, 1985). 2) The Kourovka Notebook Unsolved Problems in Group Theory, AMS Translations, series 2, Volume 121, 1983. 3) DerekJ.S. Robinson,A Course in the Theory of Groups, (Springer-Verlag, New York/Heidelberg/Berlin, 1982). 4) Joseph J. Rotman,An Introduction to the Theory of Groups (Allyn and Bacon, Inc., Boston/london/Syndney/Toronto, 1984).
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