In
mathematics, specifically
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, an abnormal subgroup is a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup ...
''H'' of a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
''G'' such that for all ''x'' in ''G'', ''x'' lies in the subgroup generated by ''H'' and ''H''
''x'', where ''H''
''x'' denotes the
conjugate subgroup
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other ...
''xHx''
−1.
Here are some facts relating abnormality to other subgroup properties:
* Every abnormal subgroup is a
self-normalizing subgroup, as well as a
contranormal subgroup.
* The only normal subgroup that is also abnormal is the whole group.
* Every abnormal subgroup is a
weakly abnormal subgroup, and every weakly abnormal subgroup is a self-normalizing subgroup.
* Every abnormal subgroup is a
pronormal subgroup, and hence a
weakly pronormal subgroup, a
paranormal subgroup, and a
polynormal subgroup.
References
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*
Subgroup properties
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