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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?

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  • $\begingroup$ See this MO-post. $\endgroup$ Mar 11 at 15:56
  • $\begingroup$ 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? $\endgroup$ Mar 11 at 18:52
  • $\begingroup$ McLain's group is a non-abelian group with trivial socle. $\endgroup$ Mar 11 at 19:19
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    $\begingroup$ What about free groups? $\endgroup$
    – Derek Holt
    Mar 11 at 22:04
  • $\begingroup$ 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. $\endgroup$ Mar 12 at 5:54

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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 $F$.

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