# Z8 is not an algebra group

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## Statement

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

## 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 $\mathbb{Z}/8\mathbb{Z}$ 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.