- #1
Samuelb88
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Homework Statement
Suppose [itex]G[/itex] contains an element of order 3, but none of order 6. Show [itex]G[/itex] is isomorphic to [itex]S_3[/itex].
Homework Equations
I am not allowed to use Sylow's theorems, or quotient groups.
The Attempt at a Solution
I've established that [itex]G[/itex] contains a subgroup [itex]H[/itex] of order 3, and three other elements of order 2. I know that [itex]H[/itex] is normal in [itex]G[/itex], while the subgroups generated by elements of order 2 are not. I also know that [itex]G[/itex] permutes the elements of order 2 by conjugation, i.e. if [itex]y \in G[/itex] is of order 2, and [itex]g \in G[/itex], then [itex]gyg^{-1}[/itex] will always have order 2.
I want to claim that that [itex]G[/itex] permutes the elements of order 3 by conjugation as well, but I am not sure if this is true, and if it is, how to prove it.
My idea is that I can somehow establish that if [itex]G[/itex] permutes the elements of order 3 by conjugation as well, then I can begin to construct an isomorphism. Unfortunately, this is just guess work, and at my point in my algebra career, I don't see how to do this and I don't have any idea how to even construct an isomorphism other than write out a multiplication table and brute force it.