Dedekind not implies abelian

From Groupprops
Revision as of 23:10, 21 January 2009 by Vipul (talk | contribs) (Dedekind not implies Abelian moved to Dedekind not implies abelian)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., Dedekind group) need not satisfy the second group property (i.e., Abelian group)
View a complete list of group property non-implications | View a complete list of group property implications
Get more facts about Dedekind group|Get more facts about Abelian group


The group property of being a Dedekind group (i.e., a group where every subgroup is normal) does not imply the group property of being an Abelian group.


Consider the quaternion group. This is a group of order eight, where every subgroup is normal. However, the group is not Abelian: the elements i and j, for instance, do not commute.

In fact, the quaternion group is in some sense the only counterexample: any non-Abelian Dedekind group is a direct product of the quaternion group and an Abelian group with the Abelian group satisfying certain conditions.