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Obraz35
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Homework Statement
Let G be a finite group. Prove that if some conjugacy class has exactly two elements then G can't be simple
Homework Equations
The Attempt at a Solution
I originally proved this accidentally assuming that G is abelian, which it isn't. So say x and y are conjugate then we have conjugacy classes {e} and {x,y}. I know that the size of a conjugacy class has to divide the order of a group so |G| is even and must be greater than 2. I'm having trouble showing the existence of a normal subgroup though.