Maximal cyclic subgroups and prime divisors in finite groups

Abstract

H e r r n P r o f e s s o r O t t o H. K e g e l z u m 7 0. G e b u r t s t a g g e w i d m e t Abstract. Let G be a finite group with no chief factor simple of Lie type E 8 (q) and C a cyclic subgroup of largest order in G. It is shown that at most two primes in the open interval ([|C|/2], |C|) divide |G|. 0. Introduction. N o t a t i o n 1. For a finite group G and a cyclic subgroup C of largest possible order in G define T := {p ∈ π(G) | [ |C| 2 ] < p < |C|}. The main goal of this paper is to prove the following Theorem: Main Theorem. Let G be a finite group with no chief factor of Lie type E 8 (q) and C a cyclic subgroup of largest order in G. Then 0 |T | 2.

Problems of Connectivity between the Sylow Graph,the Prime Graph and the Non-Commuting Graph of a Group

“…has been largely studied in [19,[38][39][40][41][42][43][44][45][46][47] and in further works of the same authors, Theorem 1.1 allows us to apply results on for investigating   G  and viceversa. In the infinite case, some variations can be discussed once appropriate restrictions are done.…”