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

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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.

Related facts

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.