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

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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.

Related facts

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.