Recently, my teacher told me about Socle of a group $G$. So, I know that the socle $Soc(G)$ of a group $G$ is the subgroup generated by all minimal normal subgroups of $G$. I was thinking then that the socle of a simple group $S$ is the group $S$ itself. What I can't find is that examples of non-abelian groups with trivial socle. After thinking a bit, I get that such a non-abelian group $G$ with trivial socle must be infinite. Also, for every normal subgroup $N$ of $G$ there exists an infinite chain of normal subgroups of $G$ inside $N$.
Can you please tell me some examples of non-abelian groups with trivial socle?
Nonabelian free groups are examples. Let $N$ be a normal subgroup of a nonabelian free group $F$. Then $N$ is free, so $N$ has proper characteristic subgroups, for example $[N,N]$ if $N$ is nonabelian and $N^2$ if $N$ is abelian (although in fact there are no nontrivial cyclic normal subgroups), so $N$ is not a minimal normal subgroup of $F$.