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Let $G$ be a finite group and let $k$ be an algebraically closed field of characteristic zero. Every $1$-dimensional representation of $G$ over $k$ factors through $G^{\mathrm{ab}} = G/[G, G]$, and every finite abelian group has a faithful representation over $k$. Taken together, these simple facts show that one can characterise the abelianisation $G^{\mathrm{ab}} = G/[G, G]$ of $G$ as the smallest quotient of $G$ such that every $1$-dimensional representation of $G$ factors through it.

Let $G_n$ be the smallest quotient of $G$ such that every $n$-dimensional representation of $G$ over $k$ factors through $G_n$. Clearly $G_1 = G^{\mathrm{ab}}$ and I've read that if $G$ has a generating set of size $m$ then $G_m = G$, i.e. $G$ has an $m$-dimensional faithful representation. What does $G_2$, or what do the $G_n$ look like in general? Can they be characterised in a different way? What do they tell us about $G$?

Martin Brandenburg
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gimothytowers
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  • By "largest quotient" do you maybe mean largest normal subgroup that you quotient by? – MPos Dec 11 '23 at 17:19
  • @MPos Thanks, I've fixed it – gimothytowers Dec 11 '23 at 17:20
  • So this essentially asks about the intersection of the kernels of reps of a given degree. While not quite the same, you may be interested in my answer https://math.stackexchange.com/a/284225/2538 which has some results about the intersection of irreps not of a given degree – Tobias Kildetoft Dec 11 '23 at 18:07
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    The post is now cross-listed and receives attention in https://mathoverflow.net/questions/460598/generalisation-of-abelianisation-using-representation-theory – MPos Dec 20 '23 at 11:16

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