By @user2345215's comment, the group $G$, defined by the presentation in question, is also defined up to isomorphism by $$\langle a, y\mid y^2, y^{-1}a^2=a^{-2}y\rangle,$$ where $y=ab$. This presentation is equivalent to $$\langle a, y\mid y^2, y^{-1}a^2y=a^{-2}\rangle,$$ which defines the gr...

There is a well-known algorithm for decomposing any given permutation as a product of (not necessarily disjoint) $2$-cycles/transpositions. Such a decomposition of a given $f\circ g$ would, in general, give strong hints about (if not completely determine) the nature of $f$ and $g$. Why? Because...

A permutation $f$ is an involution if $f\circ f=id$. As you know, any permutation can be written as a product of disjoint cycles; your permutation is $(1\ 10)(2\ 4\ 7)(3\ 5\ 8\ 6\ 9)$. In order to write an arbitrary permutation as a product of two involutions, it suffices (since disjoint permuta...