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If $m$ and $n$ are coprime, every divisor of $mn$ is uniquely $ab$ with $a\mid m$ and $b\mid n$. Hence, for a multiplicative arithmetic function $f$,

The Möbius function is $\mu (n)=0$ if a prime square divides $n$, and $\mu (n)=(-1{)}^r$ if $n$ is a product of $r$ distinct primes. The Euler totient function $\varphi (n)$ counts residues modulo $n$ coprime to $n$. From the prime factorizations,Both sides of the second identity are multiplicative, and at a prime power $p^a$,Therefore

Maschke's theorem says that every invariant subspace of a finite-dimensional complex representation of a finite group has an invariant complement. Average any projection over the group to make it equivariant. Induction on dimension therefore decomposes every representation into irreducibles.

The character of a representation is ${\chi }_V(g)=\mathrm{tr}(g{|}_V)$. For a $G$-set $X$, the permutation representation $\mathbb{C}X$ has basis $X$, and its character is the number of fixed points:For the regular action,The multiplicity of an irreducible $V_i$ is the character inner productso

If $V\cong {⨁}_i{n}_i{V}_i$, Schur lemma givesFor $V=\mathbb{C}G$, this is ${\sum }_i(\dim V_i{)}^2=|G|$.

Now suppose $\chi (g)=0$ off the identity. The multiplicity of $V_i$ in $V$ isThus $\dim V/|G|$ is a nonnegative integer and $V$ is that many copies of the regular representation. Finally,so