What is an example of a subgroup $H\le S_3$ that is not normal?

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It is not rare to see examples of subgroups of a given group, which are not normal subgroups. I'll give you one from finite groups.

Consider this finite group of order $8$, call it $G$. Then $2\in G$ and $H=\{0,1\}$ is a subgroup of $G$. But $H$ is not normal, since $$2H=\{2,4\}\ne\{2,3\}=H2.$$

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Updated on August 13, 2022

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  • snake
    snake about 1 year

    Give an example(s) of a group $G$ and $H\leq G$ (i.e., $H$ is a subgroup of $G$) where $H$ is not normal in $G$.

    What about $S_3$ and $\langle(1\;2)\rangle=\{(1),(1\;2)\}$? [Since $(1\;2\;3)(1\;2)(1\;3\;2)=(2\;3)\not\in\langle(1\;2)\rangle$] Does this work? Any other ideas?

    • David Wheeler
      David Wheeler over 8 years
      Yes, that works, and is the usual counter-example. Any such example, of course, must come from a non-abelian group, so non-abelian groups of small order are the best places to look (you can, for example, find non-normal subgroups of $D_4$ as well).