Jordan-Schur theorem on abelian normal subgroups of small index

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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: