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Q:

Invariant subgroup of a cyclic group

If $G$ is cyclic, then an invariant subgroup of $G$ is necessarily normal and $\text{Z}(G) \subseteq G’$. How is the converse of this statement true? If $G$ is abelian, the inverse is true, so I was wondering if this was still true if $G$ is not abelian.

A:

If $G$ is cyclic, then an invariant subgroup of $G$ is necessarily normal and $\text{Z}(G) \subseteq G’$.

This is not true. Take the non-abelian group of order 8, $G$:

$$G=\langle g_1, g_2, g_3 \mid g_1^2,g_2^4,g_3^8,g_2^2g_1^{ -1}g_3^2g_1^2\rangle$$
Then $\langle g_2\rangle$ is invariant under left multiplication by $g_1$ and $g_3$ but not normal. In fact, $\langle g_2\rangle$ is cyclic of order 4 and $G/\langle g_2\rangle\simeq C_2$ (which is abelian).

If $G$ is abelian, the inverse is true, so I was wondering if this was still true if $G$ is not abelian.

If $G$ is abelian, then this is true: if $A$ is a nontrivial invariant subgroup then $Z(G)\subseteq A$. Indeed, if $g\in Z(G)$ then $1=g^{ -1}g=g^{ -1}(gg^{ -1})g=(g^{ -1}g)g^{ -1}=1$ hence $g\in A$ since $A$ is invariant.
This is not true if $G$ is non-abelian. Take the dihedral group $D_8$ with presentation
$$D_8=\langle r,s \mid r^2,s^2,r^{ -1}sr,srs,srs^{ –

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