# A New Characterization of the Alternating Groups A5 and A6

### New Zealand Journal of Mathematics

Vol. 42, (2012), Pages 149-155

Alireza Khalili Asboei

Department of Mathematics,

Shariati Sari Mazandaran,

Iran

Abstract Let G be a finite centerless group, let π(G) be the set of primes p such that G contains an element of order p and let np(G) be the number of Sylow psubgroup of G, that is, np(G) = | Sylp(G) | . Set $NS(G):=\{n_{p}(G)|~p\in \pi (G)\}$. If NS(G) = NS(M), where M denotes one of the alternating simple groups A5 or A6, then $M\leq G\leq Aut(M)$.

Keywords Finite group, Sylow subgroup.

Classification (MSC2000) 20D06, 20D20.