Abelian normal not implies central
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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., abelian normal subgroup) need not satisfy the second subgroup property (i.e., central subgroup)
View a complete list of subgroup property non-implications | View a complete list of subgroup property implications
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Contents
Statement
It is possible to have a group and an abelian normal subgroup
of
(i.e.,
is an abelian group and is a normal subgroup of
) that is not a central subgroup of
(i.e.,
is not contained in the center of
).
Related facts
Similar facts
- Maximal among abelian normal implies self-centralizing in nilpotent: This shows that in a nilpotent group and in particular in a group of prime power order, any subgroup that is maximal among abelian normal subgroups is a self-centralizing subgroup. In particular, if the whole group is not abelian, it cannot be a central subgroup. Examples include dihedral group:D8, quaternion group, and many others.
- Maximal among abelian normal implies self-centralizing in supersolvable: The result also holds for supersolvable groups, such as the symmetric group of degree three.
- Normal not implies central factor
- Abelian-quotient not implies cocentral
- Nilpotent and every abelian characteristic subgroup is central implies class at most two
Opposite facts
Proof
Example of the dihedral group
Further information: dihedral group:D8, cyclic maximal subgroup of dihedral group:D8, Klein four-subgroups of dihedral group:D8
If we take to be the dihedral group of order eight, and
to be any of the three maximal subgroups of
, then
is abelian and normal in
but is not central in
.