Jordan-Schur theorem on abelian normal subgroups of small index

Statement

In terms of existence of a bounding function on index

There exists a function  such that the following holds:

Suppose  is a finite group with a faithful linear representation of degree  over the field of complex numbers (equivalently,  is isomorphic to a subgroup of the unitary group ). Then,  has an abelian normal subgroup of index at most .

Explicit description of bounding function

The smallest possible function  satisfying the above has the property that  for . The proof that this works relies on the classification of finite simple groups. For the first few values of ,  is not explicitly known for all of them. However, we do have the first few values:

  Proof or example where extreme bound is attained
1 1 Any subgroup of  is abelian by definition.
2 60 The case of special linear group:SL(2,5)

Corollary for quasirandomness

If  is a perfect group and has no proper normal subgroup of index at most , then the quasirandom degree of  is at least equal to .

Related facts

Converse

These aren't strict converses, but converse-type statements: