# Cyclic group of prime-square order is not an algebra group for odd prime

From Groupprops

This article states and (possibly) proves a fact that is true for odd-order p-groups: groups of prime power order where the underlying prime is odd. The statement is false, in general, for groups whose order is a power of two.

View other such facts for p-groups|View other such facts for finite groups

## Statement

Suppose is an odd prime number. Then, the cyclic group of prime-square order is *not* an algebra group.

## Related facts

## Facts used

## Proof

By Fact (1), if is an algebra group over , it must be isomorphic to a subgroup of . However, has exponent if is odd, so , which has exponent , cannot be isomorphic to a subgroup of it.

## References

- MathOverflow question: p-groups realisable as 1+J,where J is a nilpotent finite F-Algebra: Jack Schmidt's answer mentions this fact, and sketches a proof for the similar fact that Z8 is not an algebra group