Mathematics, 14.09.2019 02:30 ewinter13
Prove that h c g is a normal subgroup if and only if every left coset is a right coset, i. e., ah = ha for all a e g
Answers: 1
is a normal subgroup. Then by definition we must have for all 
, 
for every 
. Let 
and choose 
(
). By hypothesis we have 
, i.e. 
for some 
, thus 
. So we have 
. You can prove 
in the same way.
for all 
, we have to prove  
for every 
for some 
(by the hypothesis). hence we have 
. Because 
was chosen arbitrarily  we have the desired
.
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Prove that h c g is a normal subgroup if and only if every left coset is a right coset, i. e., ah =...
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