Dedekind 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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Definition

QUICK PHRASES: every subgroup is normal, every cyclic subgroup is normal, normal closure of element is cyclic

Symbol-free definition

A group is termed a Dedekind group or Hamiltonian group if it satisfies the following equivalent conditions:

• Every subgroup is normal.
• Every cyclic subgroup is normal.
• Every inner automorphism is a power automorphism.
• The normal closure of any singleton subset (or equivalently, the normal closure of the cyclic subgroup generated by any element) is a cyclic group.

The word Hamiltonian is reserved, by some mathematicians, for only non-abelian Dedekind groups.

Formalisms

In terms of the Levi operator

This property is obtained by applying the Levi operator to the property: cyclic group
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In terms of the Hamiltonian operator

This property is obtained by applying the Hamiltonian operator to the property: normal subgroup
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Relation with other properties

Stronger properties

• Abelian group: For proof of the implication, refer Abelian implies Dedekind and for proof of its strictness (i.e. the reverse implication being false) refer Dedekind not implies Abelian.

Weaker properties

• Group of nilpotency class two
• Nilpotent group

Metaproperties

Subgroups

This group property is subgroup-closed, viz., any subgroup of a group satisfying the property also satisfies the property
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Using the fact that normality satisfies the intermediate subgroup condition, viz any normal subgroup of the whole group is also normal in every intermediate subgroup, we conclude that any subgroup of a Dedekind group is again a Dedekind group.

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
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Using the fact that normality satisfies the image condition, viz the image of any normal subgroup via a quotient map is again a normal subgroup, we conclude that any quotient of a Dedekind group is again a Dedekind group.

Direct products

This group property is not direct product-closed. A direct product of groups with this property, need not have this property.

A direct product of two Dedekind groups need not be Dedekind. One example is the direct product of the quaternion group and the cyclic group of order four.

Testing

GAP code

One can write code to test this group property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.
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There is no GAP command to directly test whether a group is a Dedekind group, but the following GAP code can be used to define a command:

IsDedekind := function(G)
	   local H;
	   for H in List(ConjugacyClassesSubgroups(G),Representative) do
	       if not IsNormal(G,H) then return false; fi;
	       od;
	       return true;
end;;