2-Engel group: Difference between revisions
(New page: {{semistddef}} {{group property}} ==Definition== ===Symbol-free definition=== A group is termed a '''Levi group''' or a 2-'''Engel group''' if it satisfies the following equivalent ...) |
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# The normal subgroup generated by <math>x</math> is Abelian for all <math>x \in G</math>. | # The normal subgroup generated by <math>x</math> is Abelian for all <math>x \in G</math>. | ||
# The commutator <math>[x,[x,g]]</math> is the identity element for all <math>x,g \in G</math>. | # The commutator <math>[x,[x,g]]</math> is the identity element for all <math>x,g \in G</math>. | ||
==Formalisms== | |||
{{obtainedbyapplyingthe|Levi operator|Abelian group}} | |||
==Relation with other properties== | ==Relation with other properties== | ||
Revision as of 20:17, 17 October 2008
This article defines a term that has been used or referenced in a journal article or standard publication, but may not be generally accepted by the mathematical community as a standard term.[SHOW MORE]
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
Definition
Symbol-free definition
A group is termed a Levi group or a 2-Engel group if it satisfies the following equivalent conditions:
- Any two conjugate elements of the group commute.
- The normal closure of any cyclic subgroup is Abelian.
- The group is a -Engel group: the commutator between any element and its commutator with another element is the identity element.
Definition with symbols
A group is termed a Levi-group or a 2-Engel group if it satisfies the following equivalent conditions:
- commutes with for all .
- The normal subgroup generated by is Abelian for all .
- The commutator is the identity element for all .
Formalisms
In terms of the Levi operator
This property is obtained by applying the Levi operator to the property: Abelian group
View other properties obtained by applying the Levi operator