# Gaschütz group

From Groupprops

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 propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## 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

- Paper:Gaschutz54
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