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## Main Question or Discussion Point

I am reading an Abstract Algebra book, and there is a theorem that says:

Thm: If G is a finite group, then the number of elements conjugate to a is the index of the normalizer in G.

The book uses the normalizer to mean

[itex]

N(a\in G)=\{ x\in G\colon xa=ax\}

[/itex]

Now if the group is S_4, and our element is (1,2)(3,4), then the number of elements conjugate to (1,2)(3,4) is 3, and the number of elements with commute with (1,2)(3,4) is 4, so the theorem doesn't hold (???).

I am pretty sure that I am wrong, but I don't see why. Clearly the set of conjugates to (1,2)(3,4) is

[itex]

\{ (12)(34), (13)(24), (14)(23) \}

[/itex]

and the normalizer is

[itex]

\{ (12), (34), (12)(34), e\}

[/itex]

What am I missing??

Thm: If G is a finite group, then the number of elements conjugate to a is the index of the normalizer in G.

The book uses the normalizer to mean

[itex]

N(a\in G)=\{ x\in G\colon xa=ax\}

[/itex]

Now if the group is S_4, and our element is (1,2)(3,4), then the number of elements conjugate to (1,2)(3,4) is 3, and the number of elements with commute with (1,2)(3,4) is 4, so the theorem doesn't hold (???).

I am pretty sure that I am wrong, but I don't see why. Clearly the set of conjugates to (1,2)(3,4) is

[itex]

\{ (12)(34), (13)(24), (14)(23) \}

[/itex]

and the normalizer is

[itex]

\{ (12), (34), (12)(34), e\}

[/itex]

What am I missing??