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Is there a group $G$ with an element with infinite order such that every non-trivial $N \unlhd G$ contains a minimal (non-trivial) normal subgroup of $G$?

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    $\begingroup$ It seems that an infinite (non-torsion) simple group would do, but somehow that seems not very interesting. $\endgroup$ – James Jun 25 '14 at 7:02
  • $\begingroup$ yes, seems so‌. The product of 2 such groups may be a better example. thnx. $\endgroup$ – user69453 Jun 25 '14 at 7:08
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    $\begingroup$ Things like $\operatorname{SL}_n(K)$ for $K$ a field of characteristic $0$ work too (simple algebraic groups that are not simple as abstract groups). They have DCC (and ACC) on normal subgroups. $\endgroup$ – Jack Schmidt Jun 25 '14 at 19:45
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Take a field $F$. Consider the semidirect product $G$ of the additive group $A$ of $F$ by the multiplicative group $M = F^{\star}$, the latter acting on the former by multiplication.

Then $A$ is the unique minimal normal subgroup of $G$. And if the characteristic of $F$ is zero, then $G$ has elements of infinite order.

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  • $\begingroup$ $A$ is the unique minimal normal subgroup of $G$ is interesting! $\endgroup$ – user69453 Jun 25 '14 at 7:46

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