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Confbank.um.ac.irFerdowsi 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  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 . 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 ).
In this paper, we give some new results on the degree of θ-pairs and give a positive answer toConjecture 1 in  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 .
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  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  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.
 A. Ballester - Bolinches and Zho Yaoqing, on maximal subgroup of finite groups and theta pairs. Comm. in algebra, 24 (13) (1996), 4199–4209.
 J. C. Beidlemen, H. Smith, A note on supersoluble maximal subgroup and theta pair, Publ.
 P. Bhattacharya and N. P. Makherjee, On the intersection of a class of maximal subgroups of finite group II, J. Pure Appl. Algebra 42 (1986), 117–124.
A note on degree of theta pairs of finite groups  A. Erfanian and R. Rezaei, On the degree of theta pairs of finite groups, Int. J. Contemp.
Math. Sciences, Vol. 2, , no. 5, (2007) 233–239  N. P. Mukherjee, P. Bhattacharya, On theta pairs for maximal subgroup, Proc. Amer.
 Li Shirong, A note on theta pairs for maximal subgroups, Comm. in algebra, 26 (12) (1998),  Li Shirong and Z. Yaoqing, On theta pairs for maximal subgroups, J. Algebra 147 (2000)  G. Xiuyun, On theta pairs for maximal subgroup, Comm. in algebra, 22 (12) (1994), 4653–  Z. Yaoqing, On theta pairs for maximal subgroup, Comm. in algebra, 23 (6) (1995), 2099–  Z. Yaoqing, On theta pairs for maximal subgroup on finite groups, Acta Math. 40 (1997)
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