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. 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.

References

• MathOverflow question: p-groups realisable as 1+J,where J is a nilpotent finite F-Algebra

Jack Schmidt's answer sketches the above proof.