Dear all,


I am interested in residually finite, perfect groups. Are all of them known to be residually alternating? If not, how could one construct a counterexample?


A group $G$ is residually alternating if for every $g \in G$ there exists a finite alternating quotient $G/N$ such that $g \notin N$.


By a result by Katz and Magnus free groups are known to be residually alternating. This has recently been extended by Henry Wilton.


Elisabeth



Are residually finite, perfect groups residually alternating?
I think the question needs to be made a little more precise. One can take a finite simple group $G$ which is not any alternating group. Then $G$ is perfect, residually finite (!) and is not residually alternating. 


If we ask for finitely generated infinite perfect groups which are not residually alternating, then things are a little more involved: you can take $G=SL\_n({\mathbb Z})$ for $n\geq 3$. Then (by the congruence subgroup property for $SL\_n({\mathbb Z})$) the only simple quotients are $SL\_n({\mathbb Z}/p{\mathbb Z})$ for some prime $p$. These are not alternating groups if $n$ is large enough. [$SL\_n({\mathbb Z})$ is perfect, and residually finite, as can be easily seen]. 


