# Conjugate Subgroups

#### *FaerieLight*

1. Homework Statement
How would I go about proving that if K = gHg-1, for some g $\in$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.

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