# Existence of abelian normal subgroups of small prime power order

From Groupprops

## Contents

## History

This result appears to have been first noted by Burnside, in a paper published in 1912.

## Statement

Suppose is a prime number and is a finite -group of order (i.e., a group of prime power order). Then, if is a nonnegative integer such that (i.e., ), has an Abelian normal subgroup (?) (hence, an Abelian normal subgroup of group of prime power order (?)) of order .

## Particular cases

This table lists, for various values of , the smallest value of guaranteed by the theorem and the actual smallest value of such that any group of order contains an abelian normal subgroup of order .

Theorem-guaranteed | Actual smallest | Actual smallest for | |
---|---|---|---|

1 | 1 | 1 | 1 |

2 | 2 | 2 | 2 |

3 | 4 | 4 | 4 |

4 | 7 | 6 or 7 | 6 |

## Related facts

### Similar facts

### Opposite facts

- Alperin's theorem on non-existence of abelian normal subgroups of large prime power order for odd prime
- Alperin's theorem on non-existence of abelian normal subgroups of large prime power order for prime equal to two

### Corollaries

- Abelian-to-normal replacement theorem for prime-cube order
- Abelian-to-normal replacement theorem for prime-fourth order (actually, this is a corollary of the stronger version group of prime-sixth or higher order contains abelian normal subgroup of prime-fourth order for prime equal to two)

- Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime: In particular, this guarantees that for odd , and , the existence of an abelian subgroup of order implies the existence of an abelian normal subgroup of order .
- Jonah-Konvisser congruence condition on number of elementary abelian subgroups of small prime power order for odd prime
- Congruence condition on number of abelian subgroups of small prime power order and bounded exponent for odd prime
- Lower bound on order of maximal among abelian normal subgroups in terms of order of finite p-group

## Facts used

- Lower bound on order of maximal among abelian normal subgroups in terms of order of finite p-group
- Finite nilpotent implies every normal subgroup contains normal subgroups of all orders dividing its order
- Abelianness is subgroup-closed

## Proof

**Given**: A finite -group of order . .

**To prove**: has an abelian normal subgroup of order .

**Proof**:

- Let be a subgroup of that is maximal among abelian normal subgroups: Note that such a subgroup clearly exists.
- If the order of is , we have : This follows from fact (1).
- : Since , is strictly greater than , so .
- We now use Fact (2) to conclude that contains a subgroup of order that is normal in , and Fact (3) to conclude that this subgroup is abelian.

## References

### Journal references

- Paper:Burnside12
^{More info} -
*Large abelian subgroups of p-groups*by Jonathan Lazare Alperin,*Transactions of the American Mathematical Society*, Volume 117, Page 10 - 20(Year 1965):^{Official copy}^{More info}