Z8 is not an algebra group
- Algebra group is isomorphic to algebra subgroup of unitriangular matrix group of degree one more than logarithm of order to base of field size
By Fact (1), if is an algebra group over , it must be isomorphic to a subgroup of . However, has exponent 4, so , which has exponent 8, cannot be isomorphic to a subgroup of it.
- MathOverflow question: p-groups realisable as 1+J,where J is a nilpotent finite F-Algebra: Jack Schmidt's answer sketches the above proof.