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product of subgroups

For subgroups $H, K \le G$ , the set $HK = \{hk : h \in H, k \in K\}$ -- a subgroup exactly when $HK=KH$ , not automatically.

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Examples

In $S_3$ , $HK$ for two order-2 subgroups $H,K$ need not itself be a subgroup, unless $HK=KH$ .