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

Statement
Suppose $$p$$ is an odd prime number. Then, the cyclic group of prime-square order $$\mathbb{Z}/p^2\mathbb{Z}$$ is not an algebra group.

Related facts

 * Z4 is an algebra group
 * Z8 is not an algebra group

Facts used

 * 1) uses::Algebra group is isomorphic to algebra subgroup of unitriangular matrix group of degree one more than logarithm of order to base of field size

Proof
By Fact (1), if $$\mathbb{Z}/p^2\mathbb{Z}$$ is an algebra group over $$\mathbb{F}_p$$, it must be isomorphic to a subgroup of $$UT(3,p)$$. However, $$UT(3,p)$$ has exponent $$p$$ if $$p$$ is odd, so $$\mathbb{Z}/p^2\mathbb{Z}$$, which has exponent $$p^2$$, cannot be isomorphic to a subgroup of it.