Gaschütz group

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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
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Definition

A Gaschütz group is a group with the property that every subgroup in it is c-closed: in other words, every subgroup occurs as the centralizer of some subgroup.

Formalisms

In terms of the subgroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties. In other words, a group satisfies this group property if and only if every subgroup of it satisfying the first property (subgroup) satisfies the second property (c-closed subgroup), and vice versa.
View other group properties obtained in this way

The property of being a Gaschütz group can be thought of as the following two subgroup properties collapsing to the same thing in the group:

Any subgroup = c-closed subgroup

In terms of the Hamiltonian operator

This property is obtained by applying the Hamiltonian operator to the property: c-closed subgroup
View other properties obtained by applying the Hamiltonian operator

The Hamiltonian operator takes a subgroup property and outputs the property of being a group in which every subgroup has the property. The property of being Gaschütz is obtained by applying the Hamiltonian operator to the subgroup property of being c-closed.

Relation with other properties

Weaker properties

References