# Sufficiently large implies splitting for every subquotient

## Contents

## Statement

## =Verbal statement

A Sufficiently large field (?) for a finite group is a Splitting field (?) for every Subquotient (?) of the group. In particular, it is a splitting field for every subgroup as well as for every quotient group.

### Statement with symbols

Suppose is a finite group and is a sufficiently large field for . Then, if and is a normal subgroup of , then is a splitting field for the quotient group .

## Related facts

### Converse

The converse of the statement is true. In fact, if a field is a splitting field for every subgroup of a finite group, then it is sufficiently large. `For full proof, refer: Splitting field for every subgroup implies sufficiently large`

- Splitting not implies sufficiently large
- Splitting field for a group not implies splitting field for every subgroup
- Splitting field for a group implies splitting field for every quotient

## Facts used

- Sufficiently large implies splitting
- Exponent of subgroup divides exponent of group
- Exponent of quotient group divides exponent of group

## Proof

**Given**: A group , a sufficiently large field for . Subgroups with normal in .

**To prove**: is a splitting field for .

**Proof**: By facts (2) and (3), the exponent of divides the exponent of . By the definition of sufficiently large, the polynomial splits completely into distinct linear factors over , where is the exponent of . Let be the exponent of . Then the polynomial , being a factor of , also splits completely into linear factors over . (A simpler way of saying this is that a field containing primitive roots of unity also contains primitive roots of unity for ).

Thus, is sufficiently large for . Hence, by fact (1), is a splitting field for .