Question

Give an example, with justification, of a group G with subgroups H and K such
that HK is not a subgroup of G .

Answer 1

Answer on Question #81120 - Math - Abstract Algebra

Question. Give an example, with justification, of a group $G$ with subgroups $H$ and $K$ such that $HK$ is not a subgroup of $G$.

Answer. Choose $G$ to be the symmetric group $S_{4}$. Let $h=(1\,2)$, $k=(1\,3\,4)$ be elements of $G$. Choose $H$ and $K$ to be the subgroups of $G$ generated by $h$ and $k$ respectively. Clearly, $H=\{e,h\}$. We know that $h\in HK$, $k\in HK$, and $(kh)(2)=(k\circ h)(2)=3$. As $k$ fixes $2$, every element of $K$ fixes $2$. Let $h^{\prime}\in H$ and $k^{\prime}\in K$. We have that $(h^{\prime}k^{\prime})(2)=(h^{\prime}\circ k^{\prime})(2)=h^{\prime}(k^{\prime}(2))=h^{\prime}(2)$ is either $1$ or $2$. Therefore, for all $h^{\prime}\in H$ and $k^{\prime}\in K$, $kh\neq h^{\prime}k^{\prime}$. In other words, $kh\not\in HK$. This choice of $h$ and $k$ shows that $HK$ is not closed under the operation of $G$. Hence $HK$ is not a subgroup of $G$ by the definition of subgroup.

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