= Cyclic vector for a group representation
A <vector> $v$ is cyclic for a <group representation> $\rho:G\to GL(V)$ if the <linear span> of $\{\rho(g)v:g\in G\}$ is all of $V$. Equivalently, $v$ generates $V$ as a <module> over the <group algebra>. For a <finite group> over a <field> in which $|G|$ is invertible, the averaging <projection operator> $P=|G|^{-1}\sum_g\rho(g)$ has image the <fixed-point subspace of a group action>. Since $P\rho(g)v=Pv$, cyclicity implies $V^G=kPv$, so this space has <dimension> at most one.
The invertibility hypothesis matters. In <characteristic> $p$, let $G=C_p\times C_p$ and let $V=k[x,y]/(x,y)^2$, with the two generators acting by multiplication by $1+x$ and $1+y$. The <vector> $1$ is cyclic, but both $x$ and $y$ are fixed. Thus a cyclic <group representation> in <modular representation theory> can have a two-dimensional <fixed-point subspace of a group action>.
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