Non-abelian groups with trivial socle.

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?

• See this MO-post. Mar 11 at 15:56
• I am sorry, I couldn't understand the reference that you mentioned. Can you please give me some example of a non-abelian non-simple group with trivial socle? Mar 11 at 18:52
• McLain's group is a non-abelian group with trivial socle. Mar 11 at 19:19
• What about free groups? Mar 11 at 22:04
• I am not able to show that free groups do not possess minimal normal subgroups, whereas I can see that free groups do have normal subgroups. Mar 12 at 5:54

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 $$F$$.