# Congruence condition on number of elementary abelian subrings of prime-square order in nilpotent Lie ring

## Contents

## Statement

Suppose is a nilpotent Lie ring of order for some prime number . Let be the collection of subrings of of order that are abelian (i.e., the Lie bracket is trivial) with additive group the elementary abelian group of prime-square order. Then, either is empty or the size of is congruent to 1 modulo .

## Related facts

### Analogous facts for groups

- Congruence condition on number of elementary abelian subgroups of prime-square order for odd prime
- On the other hand, the analogous statement for groups
*fails*for the prime two, i.e., for the Klein four-group.

### Similar facts

- Congruence condition on number of subrings of given prime power order in nilpotent Lie ring
- Congruence condition on number of abelian subrings of prime-cube order in nilpotent Lie ring

## Facts used

- Congruence condition on number of subrings of given prime power order and bounded exponent in nilpotent ring
- Classification of Lie rings of prime-square order

## Proof

**Given**: A nilpotent Lie ring of order . is the collection of abelian subrings of that have order and whose additive group is elementary abelian.

**To prove**: Either is empty or the size of is congruent to 1 mod .

**Proof**: The proof follows directly from Fact (1), setting the bound on exponent as and noting that the only nilpotent *Lie* ring of order and exponent is the abelian Lie ring whose additive group is elementary abelian of order (by Fact (2)).