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Math Help - Algebra, Problems For Fun (37)

  1. #1
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    Algebra, Problems For Fun (37)

    Let G be a finite non-abelian group and f \in \text{Aut}(G). Let S=\{x \in G: \ f(x)=x^{-1} \}. Prove that: |S|\leq \frac{3}{4}|G|.
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  2. #2
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    What about this approach: Suppose |S|>\dfrac{3|G|}{4}

    Though S is not necessarily a subgroup, it does contain the identity and inverses by the automorphism properties, i.e. if e is the identity and x\in S, f(e)=e;\, f(x^{-1})=f(x)^{-1}=(x^{-1})^{-1}=x. Then in particular f(f(x))=x, and the set of fixed points of f\circ f does form a subgroup. Since this subgroup contains more than half of the elements it follows that f(f(x))=x for all elements of the group.

    If we can somehow show that in fact S=G, we are done.
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