Z8 is not an algebra group

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Template:Group property dissatisfaction

Statement

The group cyclic group:Z8, defined as the cyclic group of order 2^3= 8, 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}/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