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

From Groupprops

## Contents

## Statement

### Statement in terms of quasirandom degrees

Suppose is a finite group that is generated by the union of subgroups of . Then, the quasirandom degree of is bounded by the *minimum* of the quasirandom degrees of .

### Statement in terms of -quasirandom groups

Equivalently, if are all -quasirandom groups for some positive integer , then so is .

## Related facts

- Quasirandom degree of quotient group is bounded below by quasirandom degree of whole group
- Quasirandom degree of extension group is bounded below by minimum of quasirandom degrees of normal subgroup and quotient group

## 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.