Involutions are either conjugate or have an involution centralizing both of them
From Groupprops
This article gives the statement, and possibly proof, of a statement where the conclusion of the statement involves a disjunction (OR) of two possibilities. The prototypical form is: "every A is a B or a C."
Statement
Suppose is a finite group and are two involutions of . Then, at least one of these two is true:
- and are in the same conjugacy class.
- There exists an involution centralizing both and .
Facts used
Proof
Given: A finite group , two involutions of .
To prove: Either or are conjugate or there is an involution centralizing both of them.
Proof: By fact (1), is a dihedral group of order , where is the order of . This dihedral group has generating a cyclic subgroup of order and acts on this by inverting .
If is odd, then are conjugate in , hence in . Otherwise, is even, in which case the element is an involution centralizing both and .
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, Page 301, Chapter 9 (Groups of even order), Theorem 1.2, ^{More info}