Classification of finite groups with all elements of prime order

Abstract

Abstract.A finite group having all (nontrivial) elements of prime order must be a p-group of exponent p , or a nonnilpotent group of order paq , or it is isomorphic to the simple group A<.

Intersection graphs of cyclic subgroups of groups

“…Proof. It is proved in [6] that in a finite group, every element of it is of prime order if and only if it is one of p-group with exponent p or nilpotent group of order p α q or A 5 . This fact together with Theorem 2.2 completes the proof.…”

Corrigendum and addendum to: ‘‘Classification of finite groups with all elements of prime order” [Proc.\ Amer.\ Math.\ Soc.\ {\bf 106} (1989), no.\ 3, 625–629; MR0969518 (89k:20038)] by Deaconescu

“…Let &> be the class of finite groups having all (nontrivial) elements of prime order. The aim of this note is to correct some mistakes and misprints of [2] and to give a more compact description of these groups, which offers a better insight into their structure.…”

“…The statement of the Main Theorem of [2] (shortly: M.T.) is incorrect: Case 11(a) does not occur and Case Il(a') \G\ = p"q, 3 < p < q, a>3, \F(G)\ = \G'\ = pa , is missing.…”

“…is incorrect: Case 11(a) does not occur and Case Il(a') \G\ = p"q, 3 < p < q, a>3, \F(G)\ = \G'\ = pa , is missing. The omission stemmed from the incorrect application of Lemma 2.8 of [2] to Case 11(b) of the proof of the M.T.…”

“…Some misprints affect the statements of 2.8 (put H < G instead of H ^ G), of 2.10 (put K < G instead of K ^ G), and the 11th line from the bottom of page 628 of [2] (put CG(F(G)) < G instead of CG(F(G)) = G). Finally, the statements of 2.6 and 2.7 of [2] are misquoted (put Nq((x)) instead of Cq(x)) . Although these two misquoted results affect the proof of Case III of M.T., the outcome is correct: A$ is the unique nonsolvable <^-group.…”