Re: Finite Permutation Groups

mareg_at_mimosa.csv.warwick.ac.uk
Date: 01/18/05 

Date: Tue, 18 Jan 2005 16:09:58 +0000 (UTC)

In article <200501181106.j0IB69e07939@proapp.mathforum.org>,
        poinsot@univ-tln.fr (Lorenzo) writes:
>Hi!
>
>I would like to know if the two following properties are true and in this case, if anyone has a proof of them.
>
>Thanks.
>
>1) If a group G acts regularly on a finite Abelian group H then G is Abelian itself;

I presume you mean that G acts regualrly on the nonidentity elements of H ?
Is so, then the answer is no.
The smallest example is H = C_3 x C_3, G = Q_8.

>2) If a group G acts sharply 2-transitively on a finite Abelian group H then G is NONABELIAN.

This is true except for a few trivial examples, like H = C_3, G = C_2,
and H = C_2 or C_1, G = C_1.

By the way, if G acts transitively on the non-identity elements of any
finite group H, then H is abelian.

Derek Holt.