Abelian marginal subgroup
From Groupprops
This article describes a property that arises as the conjunction of a subgroup property: marginal subgroup with a group property (itself viewed as a subgroup property): abelian group
View a complete list of such conjunctions
Contents
Definition
A subgroup of a group
is termed an abelian marginal subgroup if
is an abelian group (i.e., it is an abelian subgroup of
) and
is also a marginal subgroup of
.
Examples
- The center in any group is an abelian marginal subgroup.
- In a finite p-group
, the socle coincides with
, the set of elements of order dividing
in the center (see socle equals Omega-1 of center in nilpotent p-group). This is an abelian marginal subgroup.
Examples in small finite groups
Here are some examples of subgroups in basic/important groups satisfying the property:
Here are some examples of subgroups in relatively less basic/important groups satisfying the property:
Group part | Subgroup part | Quotient part | |
---|---|---|---|
Center of dihedral group:D8 | Dihedral group:D8 | Cyclic group:Z2 | Klein four-group |
Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:
Group part | Subgroup part | Quotient part | |
---|---|---|---|
Center of M16 | M16 | Cyclic group:Z4 | Klein four-group |
Center of direct product of D8 and Z2 | Direct product of D8 and Z2 | Klein four-group | Klein four-group |
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
marginal subgroup of abelian group | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
abelian characteristic subgroup | abelian and a characteristic subgroup: invariant under all automorphisms | |FULL LIST, MORE INFO | ||
abelian normal subgroup | abelian and a normal subgroup: invariant under all inner automorphisms | |FULL LIST, MORE INFO |