Fixed-point-free involution on finite group is inverse map

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Statement

Let G be a finite group and \sigma:G \to G be an automorphism that is involutive i.e. \sigma^2 is the identity map. Suppose, further, that \sigma is fixed-point-free. Then, \sigma is the inverse map from G to itself and G is an odd-order abelian group.

Related facts

Facts used

  1. Commutator map with fixed-point-free automorphism is injective
  2. Inverse map is automorphism iff abelian
  3. Cauchy's theorem

Proof

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Given: A finite group G, a fixed-point-free involution \sigma of G

To prove: G is an abelian group and \sigma sends every element to it inverse.

Proof:

Step no. Assertion/construction Facts used Given data used Previous steps used Explanation
1 The map x \mapsto x \sigma(x^{-1}) is injective from G to itself. Fact (1) \sigma is fixed-point-free. Fact-given direct.
2 The map x \mapsto x\sigma(x^{-1}) is surjective from G to itself. G is finite Step (1) Step-given direct.
3 If a \in G is of the form a = x \sigma(x^{-1}) for some x \in G, then \sigma(a) = a^{-1}. \sigma has order two and is an automorphism We have \sigma(a) = \sigma(x\sigma(x^{-1}) = \sigma(x) \sigma^2(x^{-1} = \sigma(x)x^{-1} = (x \sigma(x^{-1}))^{-1} = a^{-1}.
4 \sigma sends every element to its inverse. Steps (2), (3) Step-combination direct.
5 G is abelian and \sigma is its inverse map. Fact (2) Step (4) Step-fact combination direct.
6 G is abelian of odd order and \sigma is its inverse map. Fact (3) Step (5) By Fact (3), if G had even order, it would have an element of order two. This would be a fixed point under \sigma, contradicting the fixed-point-free nature of \sigma. Thus, G has odd order.

References

  • Topics in Algebra by I. N. Herstein, More info, Page 70, Problems 10-11
  • Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, More info, Page 41, Exercise 23, Section 1.6 (Homomorphisms and isomorphisms)
  • Finite Groups by Daniel Gorenstein, ISBN 0821843427, More info, Page 336, Theorem 1.4, Section 10.1 (Elementary properties)