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Home / Modern Abstract Algebra / Let H and K be any two subgroups of a group G. There HK is a subgroup of G if and only if HK = KH.

Let H and K be any two subgroups of a group G. There HK is a subgroup of G if and only if HK = KH.

Proof: Let HK be a subgroup. Then (HK)-1 = HK

⟹ K-1 H-1 = HK

⟹ KH = HK [ Since H and K being subgroups H-1 = H and K-1 = K]

Conversely let HK = KH.

To prove that HK is a subgroup, we are to show that (HK) (HK)-1 = HK.

Now, (HK) (HK)-1 = (HK) (H-1 k-1)

= H(KK‑1)H-1 [by associatively]

=(HK)H-1       [since K is a subgroup, KK‑1]        

= (KH)H-1       [since HK = KH]

= K(HH-1)       [since H is a subgroup, HH‑1 = H]

= KH

= HK

Hence HK is a subgroup. (Proved)

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