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I know there is direct product $H \times K$ and semidirect product, but what is implied when it is just $HK$ without any symbols in between? For example, in this question it says $[HK:K] \leq [G:K]$. I'm guessing this means $\frac{\lvert HK \rvert}{\lvert K \rvert} \leq \frac{\lvert G \rvert}{\lvert K \rvert}$, so $\lvert HK \rvert \lvert K \rvert \leq \lvert G \rvert \lvert K \rvert$. This implies that $\lvert HK \rvert \leq \lvert G \rvert$, so either $HK = G$ or $HK$ is a subgroup of $G$. So does $HK$ just mean $H \cup K$ such that $H \cup K$ is a group (even though a union of subgroups is not always a group)?

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mr eyeglasses
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1 Answers1

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$HK$ denotes the set $\bigl\{hk\mid h\in H,k\in K\}$. In general it is not a subgroup of $G$, unless one of $H, K$ is a normal subgroup of $G$.

It is involved in the Second Isomorphism theorem:

If $H$ is a subgroup of $G$ and $K$ is a normal subgroup, then $H\cap K$ is a normal subgroup of $H$, and we have a canonical isomorphism: \begin{align*}H/(H\cap K)&\simeq HK/K\\ h(H\cap K)&\mapsto h K\end{align*}

Bernard
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