Quasirandom degree of group is bounded below by minimum of quasirandom degrees of generating subgroups

Statement

Statement in terms of quasirandom degrees

Suppose $G$ is a finite group that is generated by the union of subgroups $G_1,G_2, \dots G_k$ of $G$. Then, the quasirandom degree of $G$ is bounded by the minimum of the quasirandom degrees of $G_1,G_2,\dots,G_k$.

Statement in terms of $D$-quasirandom groups

Equivalently, if $G_1,G_2,\dots,G_k$ are all $D$-quasirandom groups for some positive integer $D$, then so is $G$.

Proof

The key idea behind the proof is to note that if a representation restricts to the trivial representation on all the generating subgroups, it must be the trivial representation on the whole group.