Group in which any two elements generating the same cyclic subgroup are automorphic
From Groupprops
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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Contents
Definition
A group in which any two elements generating the same cyclic subgroup are automorphic is a group satisfying the following equivalent conditions:
-
are such that
, then there is an automorphism of
sending
to
.
- There exists a group
containing
as a normal subgroup such that all elements of
are rational elements of
.
Relation with other properties
Stronger properties
- Group with two conjugacy classes
- Group whose automorphism group is transitive on non-identity elements
- Group in which every element is order-conjugate
- Group in which every element is order-automorphic
- Rational group
- Finite abelian group
Weaker properties
Facts
- Normal subgroup of rational group implies any two elements generating the same cyclic subgroup are automorphic
- Alternating group implies any two elements generating the same cyclic subgroup are automorphic
Metaproperties
Characteristic subgroups
This group property is characteristic subgroup-closed: any characteristic subgroup of a group with the property, also has the property
View characteristic subgroup-closed group properties]]
Quotients
This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
View a complete list of quotient-closed group properties