# Jordan-Schur theorem on abelian normal subgroups of small index

## Contents

## 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: