Cyclic subgroups

**d_p_osters** · March 9th 2009, 12:12 AM

If $X$ is a cyclic normal subgroup of $Y$ I need to prove that any subgroup of $X$ is also a normal subgroup of $Y$ .

This seems trivial but I'm not sure that it is.

**Opalg** · March 9th 2009, 02:57 AM

Originally Posted by d_p_osters

If $X$ is a cyclic normal subgroup of $Y$ I need to prove that any subgroup of $X$ is also a normal subgroup of $Y$ .

This seems trivial but I'm not sure that it is.

If X has generator a, and Z is a subgroup of X, generated by $a^k$ , then Z is the set of all elements of the form $a^{nk}$ .

If $g\in G$ , then $g^{-1}Zg$ is generated by $g^{-1}a^kg = (g^{-1}ag)^k$ , from which you can see that $g^{-1}ZG\subseteq Z$ .

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