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 $G$?
2026-03-31 08:43:42.1774946622
Minimal normal subgroups in a non-torsion group
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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.