# Group of prime-sixth or higher order contains abelian normal subgroup of prime-fourth order for prime equal to two

From Groupprops

## Statement

Any group of order contains an Abelian normal subgroup (?) (hence, Abelian normal subgroup of group of prime power order (?)) of order . In other words, for the prime , any group of order contains an abelian normal subgroup of order .

## Related facts

- Existence of abelian normal subgroups of small prime power order: This states that if , then any finite -group of order has an abelian normal subgroup of order . Note that our result does
*not*follow from this because the needed for is . - Abelian-to-normal replacement theorem for prime-cube order
- Abelian-to-normal replacement theorem for prime-fourth order