  We define a Euler characteristic $\chi(X,G)$ for a finite cell complex $X$
with a finite group $G$ acting cellularly on it. Then, each $K_{i}(X)$ (a
complex vector space with basis the $i$-cells of $X$) is a representation of
$G$, and we define $\chi(X,G)$ to be the alternating sum of the representations
$K_{i}(X)$, as elements of the representation ring $R(G)$ of $G$. By adapting
the ordinary proof that the alternating sum of the dimensions of the chain
complexes is equal to the alternating sum of the dimensions of the homology
groups, we prove that there is another definition of $\chi(X,G)$ with the
alternating sum of the representations $H_i(X)$, again as elements of the
representation ring $R(G)$. We also show that the character of this virtual
representation $\chi(X,G)$, with respect to a given element $g$, is just the
ordinary Euler characteristic of the fixed-point set by this element. Finally,
we give a topological proof of a version of Artin's induction theorem. More
precisely, we show that, if $G$ is a group with an irreducible representation
of dimension greater than 1, then each character of $G$ is a linear combination
with rational coefficients of characters induced up from characters of proper
subgroups of $G$.
