Z8 is not an algebra group

Statement
The group cyclic group:Z8, defined as the cyclic group of order $$2^3= 8$$, is not an algebra group.

Related facts

 * Z4 is an algebra group
 * Cyclic group of prime-square order is not an algebra group for odd prime

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 $$\mathbb{Z}/8\mathbb{Z}$$ is an algebra group over $$\mathbb{F}_2$$, it must be isomorphic to a subgroup of $$UT(4,p)$$. However, $$UT(4,p)$$ has exponent 4, so $$\mathbb{Z}/8\mathbb{Z}$$, which has exponent 8, cannot be isomorphic to a subgroup of it.