Semisimplity Condition and Covering Groups by Subgroups

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1 International Journal of Algebra, Vol. 4, 2010, no. 22, Semisimplity Condition and Covering roups by Subgroups Mohammad Javad Ataei Department of Mathematics Payame Noor University (PNU), Isfahan, Iran Abstract A cover for a group is a collection of proper subgroups whose union is the whole group. A cover is irredundant if no proper sub-collection is also a cover and is called maximal if all its members are maximal subgroups. For an integer n>2, a cover with n members is called an n-cover. A cover C for a group is called a C n -cover whenever C is an irredundant maximal core-free n-cover for and in this case we say that is a C n -group. Let denote a semisimple C 8 -group and {M i 1 i 8} be a maximal irredundant 8-cover for, with core-free intersection D. Also for each i, 1 i 8 we assume that : M i = α i such that α 1 α 2 α 3 α 4 α 5 α 6 α 7 α 8. In this paper we prove that if be a semisimple C 8 -group and α 3 6, then every minimal normal subgroups of isomorphic to Alt 5 or Alt 6. Mathematics Subject Classification: 20F99 Keywords: covering groups by subgroups, semisimple group, maximal iredundant cover, core-free intersection 1 Introduction and results Let be a group. A set C of proper subgroups of is called a cover for if its set-theoretic union is equal to. If the size of C is n, we call C an n-cover for the group. A cover C for a group is called irredundant if no proper subset of C is a cover for. A cover C for a group is called core-free if the intersection D = M C M of C is core-free in, i.e. D = g g 1 Dg is the trivial subgroup of. A cover C for a group is called maximal if all the members of C are maximal subgroups of. A cover C for a group is called a C n -cover whenever C is an irredundant maximal core-free n-cover for and

2 1064 M. J. Ataei in this case we say that is a C n -group. A finite group is called semisimple if it has no non-trivial normal abelian subgroups (see p. 86 of [10] for further information on such groups). Also we use the usual notations ([10]); for example, C n denotes the cyclic group of order n, (C n ) j is the direct product of j copies of C n, the core of a subgroup H of is denoted by H. In [11], Scorza determined the structure of all groups having an irredundant 3-cover with core-free intersection. Theorem 1.1 (Scorza [11]) Let {A i : 1 i 3} be an irredundant cover with core-free intersection D for a group. Then D =1and = C 2 C 2. In [8], reco characterized all groups having an irredundant 4-cover with core-free intersection. Bryce et al.[4], characterized groups with maximal irredundant 5-cover with core-free intersection. We characterized groups with maximal irredundant 6-cover with core-free intersection in [1]. Abdollahi et al.[3], characterized groups with maximal irredundant 7-cover with core-free intersection. Also we characterized p-groups with maximal irredundant 8-cover with core-free intersection in [2]. Theorem 1.2 ( See [2] ). Let be a C 8 -group. Then is a p-group for a prime number p if and only if = (C 3 ) 4 or (C 7 ) 2. Further problems of a similar nature, with slightly different aspects, have been studied by many people (see [5,6,8,11,12]). In this section we prove Proposition 1.3 and Theorem 1.4 and to do so throughout we let denote a semisimple C 8 -group and {M i 1 i 8} be a maximal irredundant 8-cover for, with core-free intersection D = 8 i=1 M i. Also for each i, 1 i 8 we assume that : M i = α i such that α 1 α 2 α 3 α 4 α 5 α 6 α 7 α 8. Proposition 1.3 Let be a semisimple C 8 -group. (a) If α 1 α 2 4, then α 3 6. (b) If α 1 α 2 4 and α 3 =6, then α i =6for 3 i 8 and also (M i ) 1for 1 i 8. Furthermore, (M i ) (M j ) (M k ) =1for 3 i<j<k 8. Theorem 1.4 Let be a semisimple C 8 -group. If α 3 6, then every minimal normal subgroups of isomorphic to Alt 5 or Alt 6.

3 Semisimplity condition and covering groups by subgroups C 8 -groups and semisimplity condition] We shall need the following lemmas Lemma 2.1 (See Lemma 2.2 of [4]) Let Γ={A i : 1 i m} be an irredundant covering of a group whose intersection of the members is D. (a) If p is a prime, x a p-element of and {i : x A i } = n, then either x D or p m n. (b) j i A j = D for all i {1, 2,..., m}. (c) If i S A i = D whenever S = n, then i T A i : D m n +1 whenever T = n 1. (d) If Γ is maximal and U is an abelian minimal normal subgroup of, then if {i : U A i } = n, either U D or U m n. Lemma 2.2 [See lemma 3.1 of [13]] Let M be a proper subgroup of the finite group and let H 1, H 2,..., H k be subgroups with : H i = β i and β 1 β 2... β k. If = M H 1... H k then β 1 k. Furthermore if β 1 = k then β 1 = β 2 =... = β k = k and H i H j M for all i j. Lemma 2.3 [See lemma 3.2 of [13]] Let N be a normal subgroup of the finite group. LetU 1,...,U h be proper subgroups of containing N and V 1,..., V k be sungroups such that V i N = with : V i = β i and β 1 β 2... β k. If = U 1... U h V 1... V k then β 1 k. Furthermore if β 1 = k then β 1 = β 2 =... = β k = k and V i V j U 1... U h for all i j. Remark 2.4 (1) The only primitive subgroups of degree 5 are C 5, C 5 C 2, C 5 C 4, Alt 5 and Sym 5. (2) The only primitive subgroups of of degree 6 are Alt 5, Alt 6, Sym 5 and Sym 6. (3) The only primitive subgroups of of degree 7 are C 7, C 7 C 2, C 7 C 3, AL(1, 7), PSL(3, 2), Alt 7 and Sym 7. Now prove the following Lemmas which will also be need in the proof of our main result. Lemma 2.5 Let be a semisimple C 8 -group. Then for every subset S of {1,...,8} such that S =4, we have i S (M i) =1. Proof 2.6 Suppose, on the contrary, K := i S (M i) 1 and S = 4, therefore K does not contain any 5-element and 7-element by Lemma 2.1 (a). Thus K is a normal soluble subgroup of, which is contradiction by semisimplity of. Lemma 2.7 Let be a semisimple C 8 -group. If α l 4 for l 8, then l i=1 (M i) 1

4 1066 M. J. Ataei Proof 2.8 If l i=1 (M i) =1then = l i=1 (M Sym 4 Sym 4. i) }{{} l then is soluble, which it is not since is semisimple. Remark 2.9 By Lemma 2.7 and Lemma2.5 we conclude that l 3 Lemma 2.10 If α 1 α 2 4 then (M i ) (M j ) =1for 3 i<j 8. Proof 2.11 Suppose, on the contrary, if (M i ) (M j ) 1for some 3 i<j 8, then there exist a minimal normal of, say U that U (M i ) (M j ) for 3 i<j 8. On the other hand we know that (M 1 ) (M 2 ) 1, then (M 1 ) (M 2 ) Sym 4 Sym 4. If R be CR-centerless radical of, then R = R n n (M 1 ) (M 2 ). Since U R, therefore U =1, which is contradiction. Lemma 2.12 Let be a semisimple C 8 -group. If α 1 α 2 4 and N := (M 1 ) (M 2 ), then N M i for 3 i 8. Proof 2.13 Suppose, on the contrary, N M j for some 3 j 8, say j. Then N (M j ). Since N (M j ) (M k ) =1, for 3 k j 8, therefore (M 1 ) (M 2 ) (M k ) =1, for 3 k j 8. Also by Lemma 2.10, we conclude that j is unique. Therefore if T := (M 1 ) (M 2 ) (M j ), then T 1. Therefore since N M i for 3 i j 8 and = NM i,by Lemma 2.3, α j 5 for 3 j j 8. Since (M 1 ) (M 2 ) (M j ) =1, for 3 j j 8. Therefore for 3 j j 8 = (M 1 ) (M 2 ) (M j ) (M 1 ) (M 2 ) (M j ). Also (M 1 ) and (M 2 ) are primitive groups of degree at most 4, and (M j ) is primitive groups of degree at most 5 for 3 j j 8. We have used the following function written with AP [7] program to all of the subdirect product of { (M 1 ), (M 2 ), (M j ) }, for 3 j j 8. The output of this program is a list of groups.

5 Semisimplity condition and covering groups by subgroups 1067 f3:=function(,h,k) local S,M,T,R,Q,W,i,j; M:=SubdirectProducts(,H); S:=[]; for i in [1..Size(M)] do Add(S,SubdirectProducts(M[i],K)); od; T:=S; R:=[]; for i in [1..Size(T)] do Add(R,T[i]); od; return R; Q:=R; W:=[]; for i in [1..Size(Q)] do Add(W,List(Q[i],x->Size(Radicalroup(x)))); od; return W;end; Now by the following command one can test each of these groups for being semisimple: Size(Radicalroup(W)) Then the group W is semisimple if and only if the output of this latter command is 1. We obtain that all of the size of radical groups are nontrivial, which is contradiction by semisimplity of. Proof Proposition 1.3 (a) If N := (M 1 ) (M 2 ), then N is non-trivial. By Lemma 2.12 N M i for 3 i 8 and = NM i, by Lemma 2.3, α 3 6. (b) If α 3 = 6 then by Lemma 2.3, α 3 = α 4 = α 5 = α 6 = α 7 = α 8 =6, M i M j M 1 M 2 for 2 i<j 8. It follow that M i M j M 1 or M 2. Thus by Lemma 2.5, (M i ) (M j ) (M k ) = 1 for 3 i<j<k 8. If for some i {3,...,8}, (M i ) = 1 then = (M i ) Sym 6. Therefore is a primitive group of degree at most 6, so by Remark 2.4 isomorphic to Alt 5 or Sym 5, Alt 6 or Sym 6, these groups have not 8-cover, which is contradiction. If for some i {1, 2}, (M i ) = 1 then cyclic group of order two or primitive group of degree at most 4. Therefore is soluble, which is contradiction by semisimplity of. Proof Theorem 1.4 If U is a minimal normal subgroup of, then U D since D = 1. Therefore there exists an index i 3 such that U (M i ) by Lemma 2.5 and so U (M i ) = 1. But := (M i ) is a primitive group of degree 6 for i 3 by Lemma 1.3(b). On the other hand has a minimal normal subgroup U := U(M i) (M i ) = U so, by semisimplicity of and Remark 2.4, U = Alt 5 or Alt 6.

6 1068 M. J. Ataei References [1] A. Abdollahi, M.J. Ataei, M. Jafarian, A. Mohammadi Hassanabadi, On groups with maximal irredundant 6-cover, Comm. Algebra, 33 (2005), [2] A. Abdollahi, M.J. Ataei, A. Mohammadi Hassanabadi, Minimal blocking set in P(n, 2) and covering groups by subgroups. Comm. Algebra, 36 (2008), [3] A. Abdollahi, M. Jafarian, A. Mohammadi Hassanabadi, On groups with an irredundant 7-cover, Journal of pure and applied algebra, 209 (2007) [4] R.A. Bryce, V. Fedri and L. Serena, Covering groups with subgroups, Bull. Austral. Math. Soc. 55 (1997), [5] R.A. Bryce, V. Fedri and L. Serena, Subgroup coverings of some linear groups, Bull. Austral. Math. Soc. 60 (1999), [6] J.H.E. Cohn, On n-sum groups, Math. Scand. 75 (1994) [7] The AP roup, AP roups, Algorithms, and Programming, Version 4.3; 2002, ( [8] D. reco, Sui gruppi che sono somma di quattro o cinque sottogruppi, Rend. Accad. Sci. Fis. Math. Napoli (4) 23 (1956), [9] B.H. Neumann, roups covered by finitely many cosets, Publ. Math. Debrecen 3 (1954), [10] D.J.S. Robinson, A course in the theory of groups, (Springer-Verlag 1982) [11]. Scorza, I gruppi che possono pensarsi come somma di tre loro sottogruppi, Boll. Un. Mat. Ital. 5 (1926) [12] M.J. Tomkinson, roups covered by finitely many cosets or subgroups, Comm. Algebra 15 (1987) [13] M.J. Tomkinson, roups as the union of proper subgroups, Math. Scand. 81 (1997), Received: June, 2010

The C 8 -Group Having Five Maximal Subgroups of Index 2 and Three of Index 3

International Journal of Algebra, Vol. 11, 2017, no. 8, 375-379 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2017.71047 The C 8 -Group Having Five Maximal Subgroups of Index 2 and Three of