Question 933859: if H and K are subgroups of a group G, then prove that HK is subgroup of G <=> HK=KH
Answer by richard1234(7193)   (Show Source):
You can put this solution on YOUR website! Suppose HK is a subgroup. Let  and  . It follows that  since we can write h = h*1, k = k*1. However, HK is closed under products, so  . Therefore  .
Also note that ^{-1} \in HK) . Then we can write ^{-1} = xy) where  ,  . This implies that ^{-1})^{-1} = (xy)^{-1} \Rightarrow hk = (xy)^{-1} = y^{-1}x^{-1} \in KH) . Therefore  , so  .
Now suppose that HK = KH. Let  , so that  ,  . Then  (by assumption), so suppose  for , . Consider the product  :
Since H and K are subgroups,  and  by closure. Therefore  .
Also note that for any  , we have
^{-1} = k_1^{-1}h_1^{-1} \in KH = HK)
Therefore elements in HK are closed under products and inverses, so HK is a subgroup.
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