1. The problem statement, all variables and given/known data
Prove that the set A= {a in G|aHa^-1=H} is a subgroup of G.
2. Relevant equations
3. The attempt at a solution
For some x,y in aHa^-1, h1,h2 in H also exists where:
X= ah1a^-1 and Y=ah2a^-1
Then, XY= ah1h2a^-1 which is contained in aHa^-1.
And x^-1= ah1^-1a^-1 which is in aHa^-1.
aHa^-1 is a subgroup of G. Since A contains a in G such that aHa^-1=H, and aHa^-1 is a subgroup of G, then A is also a subgroup of G.
Is this logic correct? Thanks!
Prove that the set A= {a in G|aHa^-1=H} is a subgroup of G.
2. Relevant equations
3. The attempt at a solution
For some x,y in aHa^-1, h1,h2 in H also exists where:
X= ah1a^-1 and Y=ah2a^-1
Then, XY= ah1h2a^-1 which is contained in aHa^-1.
And x^-1= ah1^-1a^-1 which is in aHa^-1.
aHa^-1 is a subgroup of G. Since A contains a in G such that aHa^-1=H, and aHa^-1 is a subgroup of G, then A is also a subgroup of G.
Is this logic correct? Thanks!
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