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

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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.
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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

Facts used

  1. 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.

References