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Prove or disprove the following:

$(1)$ Let $H$,$K$ be subgroups of a finite group $G$. If $|HK|=|\langle H,K \rangle|$, then $HK=\langle H,K \rangle$, that is, $HK$ is a subgroup of $G$.

$(2)$ Let $H_{1}$,$H_{2}$,..., $H_{n}$ be subgroups of a finite group $G$, where $n$ is a positive intger. If $|H_{1}H_{2}...H_{n}|=|\langle H_{1},H_{1},...,H_{n} \rangle|$, then $H_{1}H_{2}...H_{n}=\langle H_{1},H_{1},...,H_{n} \rangle$, that is, $H_{1}H_{2}...H_{n}$ is a subgroup of $G$.

My try:

$(1)$ $HK$ is a subset of $\langle H,K \rangle$. Since $|HK|=|\langle H,K \rangle|$ and $\langle H,K \rangle$ is a subgroup of $G$, then $HK=\langle H,K \rangle$.

$(2)$ Same as in $(1)$.

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  • $\begingroup$ What have you done? $\endgroup$ – Alexander Gruber Nov 20 '12 at 15:14
  • $\begingroup$ @AlexanderGruber: Please see the question again. $\endgroup$ – user28083 Nov 20 '12 at 15:38
  • $\begingroup$ Looks correct to me. Good job. $\endgroup$ – Alexander Gruber Nov 20 '12 at 16:01

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