Let $G$ be a finite solvable group, and let $G=G^{(0)}\unrhd G^{(1)}\unrhd...G^{(n)}=1$ be its derived series. Is it true that any irreducible representation of $G$ has dimension at most $n$?
2026-03-27 07:14:08.1774595648
Solvability and representation of finite groups
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The finite Heisenberg group $H_n(q)$ has $q-1$ irreducible representations of dimension $q^n$ (and $q^{2n}$ irreducible representations of dimension $1$), but its derived series just has length $2$.