# Show that a p-group has a faithful irreducible representation over $\mathbb{C}$ if it has a cyclic center

A p-group is a group of order $p^d$ where p is a prime.

If the center has order $p^m$ (since its order must divide the order of the group) then we have a one dimensional faithful irreducible representation of the center which would map a generator of the center to $e^{2\pi i/m}$. Could we then induce a representation on the whole group? If so, how do we know this is faithful and irreducible? If not, how else could we prove this?

• Inducing characters from the center behaves fairly nicely (it just gives you a multiple of the original character), and inducing a faithful character always gives a faithful character. – Tobias Kildetoft May 29 '13 at 14:13

The induced representation is faithful, but not necessarily irreducible. But, since the centre of the $p$-group $P$ is cyclic, the unique subgroup $K$ of $Z(P)$ with $|K|=p$ is contained in every nontrivial normal subgroup of $P$. Since the induced representation is faithful, at least one of its irreducible constituents does not have $K$ in its kernel, and then that consituent must be faithful.
• IThe centre $Z(P)$ is cyclic and has a faithful representation $\rho$. Then we take the induced representation $\rho_H^G$, which is a faithful representation of $G$. You will need to learn about induced representationsa if you do not know what they are. – Derek Holt Mar 17 '16 at 20:22