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]
Hence HK is a subgroup. (Proved)