Cyclic vector for a group representation
= 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>. Its projection to the invariant subspace generates that subspace, so a cyclic representation of a <finite group> has at most one copy of the <trivial representation>.