= Solution
A <group representation> is a homomorphism $D:G\to\mathrm{GL}(V)$. It is a <reducible representation> if there is a nonzero proper <invariant subspace> of $V$ preserved by every $D(g)$, and an <irreducible representation> otherwise. With $S$ invertible,
$$
\widetilde D(gh)=SD(g)D(h)S^{-1}=(SD(g)S^{-1})(SD(h)S^{-1}),
$$
so changing basis gives another representation. Its <character of a representation> is $\chi(g)=\operatorname{tr}D(g)$, and cyclicity of the <trace> gives $\widetilde\chi(g)=\operatorname{tr}(SD(g)S^{-1})=\chi(g)$.
A <conjugacy class> consists of $hgh^{-1}$ as $h$ varies. Since $(hgh^{-1})^k=hg^kh^{-1}$, a power equals the identity exactly when $g^k$ does. Thus <conjugation preserves element order>, including infinite order.
For the square, let $r$ be rotation by $\pi/2$ and $s$ reflection in a horizontal symmetry axis. The <dihedral group> has the eight elements
$$
e,r,r^2,r^3,s,rs,r^2s,r^3s,\qquad r^4=s^2=e,\quad srs=r^{-1}.
$$
The <conjugacy classes of a finite dihedral group> specialize to
$$
\boxed{\{e\},\quad\{r^2\},\quad\{r,r^3\},\quad\{s,r^2s\},\quad\{rs,r^3s\}.}
$$
The two central rotations are singletons. Reflection conjugation reverses rotations, while conjugation by $r$ shifts a reflection exponent by two, giving the two reflection classes.
For finite groups over $\mathbb C$, <character orthogonality> and completeness make the <irreducible characters> a basis of the <class functions>. The latter space has one independent value per conjugacy class. Hence \b[the square group has five inequivalent irreducible complex representations]. The <sum of squares of irreducible representation dimensions> is eight, forcing four one-dimensional representations and one two-dimensional representation. The four scalar choices $r\mapsto\pm1$, $s\mapsto\pm1$ supply the former; the ordinary action on the plane supplies the latter.
Back to article page
