Past exam of the mathematics course of the University of Cambridge / 2007 / ia / Paper 3 / 2D (Algebra and Geometry)

A group isomorphism $f:G\to H$ is a bijection satisfying $f(xy)=f(x)f(y)$ for all $x,y\in G$. It sends the identity to the identity and satisfies $f(x^k)=f(x{)}^k$. Applying its inverse shows that it preserves the order of a group element exactly.

The cyclic group $C_8$ has an element of order eight. Every element of the direct product of groups $C_4\times C_2$ has fourth power equal to the identity, and a generator of the first factor paired with the identity of the second has order four. Every nonidentity element of $C_2\times C_2\times C_2$ has order two. Thus their maximum element orders are respectively $8,4,2$, which an isomorphism would preserve. The three groups are pairwise nonisomorphic.

For a fourth example take the dihedral group of symmetries of a square. It consists of four rotations and four reflections, so has order eight. With $r$ a quarter-turn and $s$ a reflection, $srs=r^{-1}\ne r$, showing $sr\ne rs$. This group is nonabelian, whereas all three displayed direct products of groups are abelian groups. Since a group isomorphism preserves commutativity, the square's dihedral group is isomorphic to none of them. Its order is eight regardless of whether a notation convention calls it $D_4$ or $D_8$.