Bound on the size of a group given a faithful irrep of a certain dimension - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T16:53:04Z http://mathoverflow.net/feeds/question/108668 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108668/bound-on-the-size-of-a-group-given-a-faithful-irrep-of-a-certain-dimension Bound on the size of a group given a faithful irrep of a certain dimension Alonso 2012-10-02T21:15:12Z 2012-10-02T23:25:14Z <p>Let $G$ be a finite group with faithful irreducible representation $\gamma: G \to GL_n(\mathbb{C})$, $n>1$.</p> <p>Can we put a bound on the size of $G$? What if $G$ is nilpotent?</p> http://mathoverflow.net/questions/108668/bound-on-the-size-of-a-group-given-a-faithful-irrep-of-a-certain-dimension/108674#108674 Answer by Geoff Robinson for Bound on the size of a group given a faithful irrep of a certain dimension Geoff Robinson 2012-10-02T23:25:14Z 2012-10-02T23:25:14Z <p>Jordan's theorem says that there is a function $f : \mathbb{N} \to \mathbb{N}$ such that whenever $G$ is a finite subgroup of ${\rm GL}(n,\mathbb{C}),$ there is an Abelian onrmal subgroup $A$ such that $[G:A] \leq f(n)$. Explicit bounds were given later, which can be much improved by invoking the classification of finite simple groups. If $G$ is a primitive subgroup of ${\rm SL}(n,\mathbb{C})$, then it follows from Jordan's theorem that $|G|$ is bounded in terms of $n$. Recall that a primitive representation is an irreducible one which is not equivalent to a representation induced from a representation of a proper subgroup.</p>