Fixed-point-free involution on finite group is inverse map
From Groupprops
Statement
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.
Related facts
- 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.
Facts used
- Commutator map with fixed-point-free automorphism is injective
- Inverse map is automorphism iff abelian
- Cauchy's theorem
Proof
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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.
Proof:
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. |
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)