# 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 $K:\mathbb{N} \to \mathbb{N}$ such that the following holds:

Suppose $G$ is a finite group with a faithful linear representation of degree $d$ over the field of complex numbers (equivalently, $G$ is isomorphic to a subgroup of the unitary group $U_d(\mathbb{C})$). Then, $G$ has an abelian normal subgroup of index at most $K(d)$.

### Explicit description of bounding function

The smallest possible function $K$ satisfying the above has the property that $K(d) = (d + 1)!$ for $d \ge 71$. The proof that this works relies on the classification of finite simple groups. For the first few values of $d$, $K(d)$ is not explicitly known for all of them. However, we do have the first few values:

$d$ $K(d)$ Proof or example where extreme bound is attained
1 1 Any subgroup of $U_1(\mathbb{C})$ is abelian by definition.
2 60 The case of special linear group:SL(2,5)

### Corollary for quasirandomness

If $G$ is a perfect group and has no proper normal subgroup of index at most $K(d - 1)$, then the quasirandom degree of $G$ is at least equal to $d$.

## Related facts

### Converse

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