Past exam of the mathematics course of the University of Oxford / 2021 / Prelims / Mathematics I / 5 / a / i / Solution

The function composition of two bijections $f,g:X\to X$ is a bijection, with inverse function $(f\circ g{)}^{-1}=g^{-1}\circ f^{-1}$. Thus function composition is a binary operation on $\mathrm{Sym}(X)$. It is associative, since both $(f\circ g)\circ h$ and $f\circ (g\circ h)$ send $x$ to $f(g(h(x)))$.

The identity function ${\mathrm{id}}_X$ is a bijection and satisfies $f\circ {\mathrm{id}}_X={\mathrm{id}}_X\circ f=f$. Every bijection $f$ has an inverse function $f^{-1}:X\to X$, itself a bijection, satisfying $f\circ f^{-1}=f^{-1}\circ f={\mathrm{id}}_X$. These verify all group axioms, so $\mathrm{Sym}(X)$ is a group, the symmetric group of $X$.