Group representation

Remarks

Example: let $G = \Z / n\Z$ and $V = \C$. The map $\rho(k) = e^{2\pi i k / n}$, viewed as multiplication on $\C$, is a representation of degree $1$, because $e^{2\pi i (k + l)/n} = e^{2\pi i k/n} e^{2\pi i l/n}$, and the value does not depend on the representative $k$ of its class. Non-example: on $G = \Z / 2\Z$, the map sending $0$ to $\id$ and $1$ to $2 \cdot \id$ on $\C$ is not a representation, because $\rho(1 + 1) = \rho(0) = \id$ but $\rho(1)\rho(1) = 4 \cdot \id$. When $V = F^n$, the group $GL(V)$ is the general linear group $GL_n(F)$, and a representation is a matrix-valued homomorphism. Equivalently, a representation is a linear action of $G$ on $V$, or a module over the group algebra $F[G]$.