Classification of Finite Groups with a CC-Subgroup

Abstract

A proper subgroup M of a group G is called a CC-subgroup of G if the centralizer C G ðmÞ of every m 2 M # ¼ M nf1g is contained in M. In this paper we classify all finite groups containing a CC-subgroup, extending work of many authors.

Maximal cyclic subgroups and prime divisors in finite groups

“…The value of |C| is available from [2]. Then use Table 6 in [4] in order to determine primes l dividing the order of G and satisfying [|C|/2] < l < |C|.…”

“…P r o o f o f t h e C l a i m. One has G ∼ = PSL (3,4), and it has order 2 6 × 3 2 × 5 × 7. According to Table 2 cyclic groups can have order dividing either of 3 2 , 7 and 2 6 .…”

“…According to Table 2 cyclic groups can have order dividing either of 3 2 , 7 and 2 6 . The 2-Sylow of PSL (3,4) is isomorphic to the one of SL (3,4) which in turn is isomorphic to the group of all upper triangular 3 × 3 matrices, entries from GF(2)[x]/(x 2 + x + 1) and diagonal entries equal to 1. As known, every cyclic subgroup has then order at most 4.…”

“…Therefore Theorem A in [4] implies the existence of a normal series Maximal cyclic subgroups and prime divisors in finite groups 35 CC-subgroup of G/N lifts to a CC-subgroup of G, the almost simple group G/N has a prime graph with at least 4 components. In light of [11], [12] such G/N can have at most 3 prime graph components.…”

“…During the proof of Theorem 3 we shall need Table 2 which is identical to Table 5 in [4]. With s(G) we denote the number of prime graph components.…”