One-headed group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
This is a variation of simplicity|Find other variations of simplicity | Read a survey article on varying simplicity

Definition

Symbol-free definition

A group is said to be one-headed if it has a proper normal subgroup that contains every proper normal subgroup. Note that such a proper normal subgroup must also therefore be the unique maximal normal subgroup. The quotient of the group by this maximal normal subgroup is termed the head of the group.

Note that simply saying that there is a unique maximal normal subgroup is a somewhat weaker statement, though it is equivalent for a group in which every proper subgroup is contained in a maximal subgroup.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Simple group nontrivial, no proper nontrivial normal subgroup |FULL LIST, MORE INFO
Composition series-unique group has a unique composition series |FULL LIST, MORE INFO
Quasisimple group perfect group whose inner automorphism group is a simple group |FULL LIST, MORE INFO

Related properties

  • Monolithic group: This has a unique nontrivial normal subgroup contained in all the nontrivial normal subgroups.

Analogues in other algebraic structures

  • Local ring in the theory of commutative unital rings.
  • Local ring in the theory of noncommutative unital rings.