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Conjugate Subgroups

1. Homework Statement
How would I go about proving that if K = gHg-1, for some g [itex]\in[/itex]G, where K and H are both subgroups of G with a prime number of elements, then K = H?


2. Homework Equations
I've tried to prove it by saying that if K = gHg-1 then Kg = gH, and since H = gHg-1, then Hg = gH also, so Hg = Kg, and hence H = K. I don't think that this proof is valid, unfortunately. And I've just realised that H does not necessarily equal gHg-1 unless g is in the normaliser of H. :(

3. The Attempt at a Solution
This is actually something which I am attempting to prove in order to prove something else, so I'm not even sure if what I'm trying to prove holds at all. I just need K = H for my proof to work.
 

Dick

Science Advisor
Homework Helper
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Take G=S3, the symmetric group on three elements. Take K={e,(12)} and H={e,(13)}. The subgroups both have order 2 (a prime) and they are conjugate via g=(23), but they aren't equal.
 

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