Dedekind not implies abelian

From Groupprops
Revision as of 16:49, 7 July 2008 by Vipul (talk | contribs) (New page: {{subgroup property non-implication}} ==Statement== The group property of being a Dedekind group (i.e., a group where ''every'' subgroup is normal) does ''not...)
(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 subgroup properties. That is, it states that every subgroup satisfying the first subgroup property need not satisfy the second subgroup property
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
|
Property "Page" (as page type) with input value "{{{stronger}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
Property "Page" (as page type) with input value "{{{weaker}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property {{{stronger}}} but not {{{weaker}}}
  • Property "Satisfies property" (as page type) with input value "{{{stronger}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
  • Property "Dissatisfies property" (as page type) with input value "{{{weaker}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
|View examples of subgroups satisfying property {{{stronger}}} and {{{weaker}}}
  • Property "Satisfies property" (as page type) with input value "{{{stronger}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
  • Property "Satisfies property" (as page type) with input value "{{{weaker}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.

Statement

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.

Proof

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.