SOLUTION: if H and K are subgroups of a group G, then prove that HK is subgroup of G <=> HK=KH
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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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