Dimension of faithful irreducible representations of \$\mathbb{Z}_q\rtimes \mathbb{Z}_{p^2}\$ in characteristic p,q - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T23:20:14Z http://mathoverflow.net/feeds/question/118543 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118543/dimension-of-faithful-irreducible-representations-of-mathbbz-q-rtimes-mathbb Dimension of faithful irreducible representations of \$\mathbb{Z}_q\rtimes \mathbb{Z}_{p^2}\$ in characteristic p,q A.B. 2013-01-10T15:42:24Z 2013-01-10T16:56:36Z <p>Let \$p,q\$ be primes s.t. \$q=np+1\$. Denote \$m=p^2\$. Then \$\mathbb{Z}_p\$ acts non-trivially on \$\mathbb{Z}_q\$, so we have a non-abelian semi-direct product \$\mathbb{Z}_q\rtimes \mathbb{Z}_m\$, with the action of \$\mathbb{Z}_p\$. This group has a faithful irreducible representation.</p> <p>Over \$\mathbb{C}\$, every faithful irreducible representation must be of dimension \$p\$, because the dimension must divide \$p^2q\$, it cannot be 1 since the group is non-abelian, and it cannot be greater than \$p\$ because there is an abelian subgroup of order \$pq\$.</p> <p>Is it possible to bound the dimensions of the faithful irreducible representations in characteristic \$p,q\$ as well?</p> <p>Also, is there a simple way to check if the faithful irreducible representation I have is also irreducible in characteristic \$p,q\$?</p>