Fixed-point-free involution on finite group is inverse map
Let be a finite group and be an automorphism that is involutive i.e. is the identity map. Suppose, further, that is fixed-point-free. Then, is the inverse map from to itself and is an odd-order abelian group.
- Semidirect product of finite group by fixed-point-free automorphism implies all elements in its coset have same order
- Frobenius conjecture: This is a generalization of sorts of the above result. It states that if a finite group possesses a fixed point-free automorphism of prime order, the finite group must be nilpotent. This was proved by Thompson in 1959.
- Commutator map with fixed-point-free automorphism is injective
- Inverse map is automorphism iff abelian
- Cauchy's theorem
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Given: A finite group , a fixed-point-free involution of
To prove: is an abelian group and sends every element to it inverse.
|Step no.||Assertion/construction||Facts used||Given data used||Previous steps used||Explanation|
|1||The map is injective from to itself.||Fact (1)||is fixed-point-free.||Fact-given direct.|
|2||The map is surjective from to itself.||is finite||Step (1)||Step-given direct.|
|3||If is of the form for some , then .||has order two and is an automorphism||We have .|
|4||sends every element to its inverse.||Steps (2), (3)||Step-combination direct.|
|5||is abelian and is its inverse map.||Fact (2)||Step (4)||Step-fact combination direct.|
|6||is abelian of odd order and is its inverse map.||Fact (3)||Step (5)||By Fact (3), if had even order, it would have an element of order two. This would be a fixed point under , contradicting the fixed-point-free nature of . Thus, has odd order.|
- 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)