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

## Statement

### In terms of quasirandom degree

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

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

### In terms of $D$-quasirandomness

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

## Proof

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

Given: A finite group $G$, normal subgroup $N$. A nontrivial linear representation $\varphi$ of $G$ of degree $d$.

To prove: Either $N$ or $G/N$ has a nontrivial linear representation of degree $d$.

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