I'm working on a group theory exercise in which I have to show that any function in $S(X)$, the group of all bijective functions mapping $X$ to itself, can be written as the composition of two functions of order $2$, that is, functions being their own inverse. Here is $X$ is assumed to be an infinite set. In the first part I managed to show this is true for the group of congruences of $\mathbb{R}^2$ mapping the origin onto itself. However, I have no idea what to do here, since I dit not manage to find any decomposition that seems to work. Any help/hint is much appreciated!
Asked
Active
Viewed 56 times
0
-
So $;S(X);$ is not the bijective functions with finite support, but rather all the bijective functions $;X\to X;$ . – DonAntonio Feb 24 '18 at 18:56
-
Yes indeed, that is just the definition of $S(X)$ right? – Václav Mordvinov Feb 24 '18 at 18:59
-
related question – lulu Feb 24 '18 at 19:01