Quasirandom degree

Definition for finite groups
Suppose $$G$$ is a finite group. The quasirandom degree of $$G$$ is defined in the following equivalent ways:


 * 1) It is the minimum possible degree of a nontrivial linear representation of $$G$$ over the field of complex numbers.
 * 2) It is the minimum possible degree of an nontrivial defining ingredient::irreducible linear representation of $$G$$ over the field of complex numbers (i.e., it is the smallest of the defining ingredient::degrees of irreducible representations once we throw out the trivial representation).
 * 3) It is the minimum of the degrees of nontrivial linear representations of $$G$$ over all possible fields of characteristic zero.
 * 4) It is the minimum possible degree of an nontrivial defining ingredient::irreducible linear representation of $$G$$ over all possible fields of characteristic zero.

Note that for the trivial group, the quasirandom degree is taken to be $$+\infty$$.

We say that a nontrivial finite group $$G$$ is $$d$$-quasirandom if the quasirandom degree of $$G$$ is at least $$d$$, i.e., if every nontrivial irreducible linear representation of $$G$$ has degree at least $$d$$.

Quasirandom degree of one

 * For any nontrivial finite abelian group, the quasirandom degree is 1. This follows from the fact that abelian implies every irreducible representation is one-dimensional
 * More generally, the quasirandom degree of a group is greater than 1 if and only if the group is a perfect group. This follows from the fact that number of one-dimensional representations equals order of abelianization, so if the abelianization is nontrivial, there is more than one one-dimensional representation, and hence a nontrivial one-dimensional representation.

Relation with subgroups and quotients

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