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Ferdowsi University of Mashhad, March 10-12, 2010, p. 29-32 A note on degree of theta pairs of finite groups Department of Mathematics, Faculty of Sciences, Malayer University, Malayer, Iran.
Department of Mathematics, Faculty of Sciences, Malayer University, Malayer, Iran.
Let M be a maximal subgroup of a finite group G. A pair of subgroups (C, D) of G is calleda θ-pair of M if it satisfies the following conditions: (i) D < C and D = M and (iii) C/D has no proper normal subgroup of G/D. In this paper, we introduce the degree of maximal θ-pairs denoted by dθm(G) as the ratio |θm(G)|/|G|, whereθm(G) is the set of all maximal θ-pairs of the maximal subgroups of G and m(G) is totalnumber of distinct maximal subgroups of G. Moreover, we obtain some result on the degreeof θ-pairs and solvability and nilpotancy of finite groups.
Keyword and phrases: θ-pair, maximal θ-pair, degree of θ-pairs AMS subject Classification 2010: Primary: 20D10, 20E34; In [5] Mukhrejee and Bhattacharya introduced the concept of θ-pairs for maximal subgroups of afinite group and used this concept to investigate the structure of some finite groups. Beidlemanand Smith generalized the concept of θ-pairs for infinite groups in [2]. The investigations onθ-pairs are continued in [1, 8, 9, 10]. A lot of research on θ-pairs has shown that the conceptoffers a good service for studying the structure of finite groups.
In 2007 Erfanian and Rezaei introduced a degree on the θ-pairs of a finite group which was calledthe degree of θ-pairs of G, and they proved some elementary results on the concept (see [4]).
In this paper, we give some new results on the degree of θ-pairs and give a positive answer toConjecture 1 in [4] when G is a finite group with non-trivial Frattini subgroup and G/φ(G) is notsimple. Throughout this paper, all groups are assumed to be finite and all unexplained notationand terminology are standard. In addition, if H is a subgroup of G, write HG = CoreG(H), thecore of H in G.
We begin with the definition of θ-pair given by Mukhrejee and Bhattocharya in [5].
Definition 2.1 Let G be a finite group and M be a maximal subgroup of G. A θ-pair of M isany pair (C, D) of subgroups G satisfying the following conditions:(a) D (b) M, C = G and M, D = M .
(c) C/D has no proper normal subgroup of G/D. The set of all θ-pairs of a maximal subgroupM of G is denoted by θ(M ). We also denote θ(G) as the union of θ(M ) for all maximal subgroupsM of G. A partial order ≤ may be defined on θ(M ) as follows: (C, D) ≤ (C1, D1) if and only if C ≤ C1.
A note on degree of theta pairs of finite groups No condition is placed on the second component of the pairs, but if (C, D) ≤ (C1, D1), thenC ≤ C1 and so D1/D1 ≤ DD1/D1 ≤ C1/D1. By the third part of definition of θ-pair, eitherDD1 = C1 and so C1 ⊆ M which is a contradiction, or DD1 = D1 and so D ≤ D1. Furthermore, (C, D) = (C1, D1) if and only if C = C1, D = D1.
Definition 2.2 Let M be a maximal subgroup of group G. Then (C, D) is said to be a maximalθ-pair, if there is no θ-pair (C1, D1) such that C < C1.
The set of all maximal θ-pairs of a maximal subgroup M of group G is denoted by θm(M ).
We also denote θm(G) as the union of θm(M ) for all maximal subgroups M of G.
For a maximal subgroup M of any group G, (C, MG) is a maximal θ-pair in θ(M ), where C/MGis a chief factor of G. Because, MG < C, MG G and C, MG = G. Furthermore, if there is (C1, MG) ∈ θ(M ) such that (C, MG) < (C1, MG), then C < C1 and so C/MG < C1/MG whichis contradiction. Therefore, (C, MG) ∈ θm(M ).
Proposition 2.3 A group G is nilpotent if and only if for any two distinct maximal subgroupsX and Y of G, θm(X) ∩ θm(Y ) = ∅.
In [4] Erfanian and Rezaei introduced some degree on θ-pairs. In the this section we will recall the degree of maximal θ-pairs and we will state some new results on the concept.
Definition 3.1 Let G be a finite group. We define the degree of maximal θ-pairs of G as theratio where m(G) is the total number of distinct maximal subgroup of G.
Proposition 3.2 Let G/Φ(G) be a finite simple group. Then dθm(G) = 1/|m(G)|.
It must be noted that if G/Φ(G) is a non-abelian simple group, then m(G) > 1 and so by the above proposition dθm(G) < 1.
Example 3.3 Let G = a, b | an = b2 = (ab)2 = 1 be the dihedral group of order 2n such thatn = pα1 pα2 .pαk is the prime factorization of n > 1.
It is easy to verify that all maximal subgroups of G are a and api , ajb for 1 ≤ i ≤ k and0 ≤ j ≤ pi − 1. Therefore Now, let 2|n and p1 = 2α1. Then a , a2, b and a2, ab are the normal subgroups of G, becausetheir indexes in G are 2. Therefore, (G, a ), (G, a2, b ) and (G, a2, ab ) are the unique maximalθ-pairs of θm( a ), θm( a2, b ) and θm( a2, ab ) respectively. Furthermore, since for 2 ≤ i ≤ k wehave api , b aib = api , a2ib , then the maximal subgroups api , b , api , ab , . , api , api−1b areconjugate, and so they are not normal subgroups of G. By the following table we can compute the maximal θ-pairs of these maximal subgroups: ( api , ab , api ), ( api , a2b , api ), · · · ( api , b , api ), ( api , a2b , api ), · · · ( api , b , api ), ( api , ab , api ), · · · Hence, for all i ≥ 2 there are pi + 1 different maximal θ-pairs and so |θm(G)| = 1 + p1 + (p2 + 1) + · · · + (pk + 1) = k + Now assume that 2 n. Then a is the unique normal maximal subgroup G and so by the samecomputations of the last case, we have |θm(G)| = k + 1 + It must be noted that in the dihedral group G of order 2n, n = 2α if and only if dθm(G) = 1.
Lemma 3.4 Assume that N is a normal subgroup of group G such that lies in Frattini subgroupof G. Then dθm(G/N ) ≤ dθm(G) Suppose that M is a non-normal maximal subgroup of group G and G/Φ(G) is not simple.
Then there is element g ∈ G such that M = M g. It is easy to check that (M g, MG) is maximalθ-pair for M and (M, MG) is maximal θ-pair for M g. This follows us to state the followingtechnical Lemma.
Lemma 3.5 Assume that G is a group such that G/Φ(G) is not simple. Then dθm(G) ≥ 1.
Now we are ready to give a positive answer to conjecture 1 in [4] when G is a group with non-trivial Frattini subgroup and G/Φ(G) is not a simple group.
Theorem 3.6 Let G be a group with non-trivial Frattini Subgroup and G/Φ(G) be non-simple.
Then G is nilpotent if and only if dθm(G) = 1.
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