Let's consider s representation $\rho: G \rightarrow GL(V)$ such that there is a bijection between every linear operator of the image of $\rho$ and the elements of $G$. Is the image of $\rho$ isomorphic to $G$?
2026-04-06 04:55:08.1775451308
Is this representation isomorphic to the group?
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No. Consider the representation $\rho:GL(n,\Bbb C)\times \Bbb C^*\to GL(n,\Bbb C)$ sending $(A,z)\mapsto zI_n$. $GL(n,\Bbb C)\times\Bbb C^*$ and the image of $\rho$ have the cardinality of $\Bbb C$, but the former is not abelian when $n>1$.