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
This proof uses a tabular format for presentation. Provide feedback on tabular proof formats in a survey (opens in new window/tab) | Learn more about tabular proof formats|View all pages on facts with proofs in tabular format
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 ![]() ![]() |
Fact (1) | ![]() |
Fact-given direct. | |
2 | The map ![]() ![]() |
![]() |
Step (1) | Step-given direct. | |
3 | If ![]() ![]() ![]() ![]() |
![]() |
We have ![]() | ||
4 | ![]() |
Steps (2), (3) | Step-combination direct. | ||
5 | ![]() ![]() |
Fact (2) | Step (4) | Step-fact combination direct. | |
6 | ![]() ![]() |
Fact (3) | Step (5) | By Fact (3), if ![]() ![]() ![]() ![]() |
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)