INTERNATIONAL JOURNAL OF GROUP THEORY
Year:2019 | Volume:8 | Issue:1|Papers Count:5

In this paper we show that every  nite nonabelian p-group G in which the Frattini subgroup  (G) has order  p5 admits a noninner automorphism of order p leaving the center Z(G) elementwise  xed. As a consequence it follows that the order of a possible counterexample to the conjecture of Berkovich is at least p8.

If G is a  nite group and X a conjugacy class of elements of G, then we de ne rank(G: X) to be the minimum number of elements of X generating G. In the present article, we determine the ranks for the Fischer's simple group Fi ′ 24 and the baby monster group B.

For a  nite group H, let cs(H) denote the set of non-trivial conjugacy class sizes of H and OC(H) be the set of the order components of H. In this paper, we show that if S is a  nite simple group with the disconnected prime graph and G is a  nite group such that cs(S) = cs(G), then jSj = jG=Z(G)j and OC(S) = OC(G=Z(G)). In particular, we show that for some  nite simple group S, G  = S  Z(G).

For a  nite group G and a positive integer n, let G(n) be the set of all elements in G such that xn = 1. The groups G and H are said to be of the same (order) type if jG(n)j = jH(n)j, for all n. The main aim of this paper is to show that if G is a  nite group of the same type as Suzuki groups Sz(q), where q = 22m+1  8, then G is isomorphic to Sz(q). This addresses to the well-known J. G. Thompson's problem (1987) for simple groups.

A subset B of a group G is called a difference basis of G if each element g 2 G can be written as the difference g = ab 􀀀 1 of some elements a; b 2 B. The smallest cardinality jBj of a difference basis B  G is called the difference size of G and is denoted by Δ [G]. The fraction g[G]: = Δ [G]= √ jGj is called the difference characteristic of G. We prove that for every n 2 N the dihedral group D2n of order 2n has the difference characteristic p 2  g[D2n]  p48 586  1: 983. Moreover, if n  2
1015, then g[D2n] < p4 6  1: 633. Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality  80.