International Journal of Group Theory
Volume:8 Issue: 1, Mar 2019

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

If Gis a finite group and X a conjugacy class of‎ ‎elements of G‎, ‎then we define \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 finite 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 finite simple group with the disconnected prime graph and G is a finite group such that cs(S)=cs(G)‎, ‎then |S|=|G/Z(G)| and OC(S)=OC(G/Z(G))‎. ‎In particular‎, ‎we show that for some finite simple group S‎, ‎G≅S×Z(G)

‎For a finite 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 |G(n)|=|H(n)|‎, ‎for all n‎. ‎The main aim of this paper is to show that if G is a finite 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 {\em‎ ‎difference basis} of G if each element g∈G can be written as the‎ ‎difference g=ab−1 of some elements a,b∈B‎. ‎The smallest‎ ‎cardinality |B| of a difference basis B⊂G is called the {\em‎ ‎difference size} of G and is denoted by Δ[G]‎. ‎The fraction ‎‎‎ð[G]:=Δ[G]/|G|−−−√ is called the {\em difference characteristic} of G‎. ‎We prove that for every n∈N the dihedral group‎ ‎D2n of order 2n has the difference characteristic‎ ‎2–√≤ð[D2n]≤48586√≈1.983‎. ‎Moreover‎, ‎if n≥2⋅1015‎, ‎then ð[D2n]<46√≈1.633‎. ‎Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality ≤80.