Z8 is not an algebra group

From Groupprops
Revision as of 22:03, 16 August 2012 by Vipul (talk | contribs)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

Template:Group property dissatisfaction


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


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.