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

## Contents

## Statement

### In terms of quasirandom degree

Suppose is a finite group and is a normal subgroup of . Then, the quasirandom degree of is at *least* equal to the *minimum* of the quasirandom degrees of and of the quotient group .

The quasirandom degree is the minimum possible degree of a nontrivial linear representation over the field of complex numbers.

### In terms of -quasirandomness

Suppose is a finite group and is a normal subgroup of . If is a positive integer such that both and are -quasirandom, then is also -quasirandom.

## Related facts

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

## Proof

It suffices to show that if has a nontrivial linear representation of degree over the field of complex numbers, then either or has a nontrivial linear representation of degree .

**Given**: A finite group , normal subgroup . A nontrivial linear representation of of degree .

**To prove**: Either or has a nontrivial linear representation of degree .

**Proof**: Consider the restriction of to . This is a linear representation of of degree . If this is nontrivial, we are done. If it is trivial, descends to a linear representation of of the same degree , which must be nontrivial because itself is nontrivial. In that case, we are done too.