mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, 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, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

''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 iden ...

''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 ''xHx''⁻¹. Here are some facts relating abnormality to other subgroup properties: * Every abnormal subgroup is a

self-normalizing subgroup In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...

, 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 In mathematics, especially in the field of group theory, a pronormal subgroup is a subgroup that is embedded in a nice way. Pronormality is a simultaneous generalization of both normal subgroups and abnormal subgroups such as Sylow subgroups, . A ...

, and hence a weakly pronormal subgroup, a paranormal subgroup, and a polynormal subgroup.

References

* * * * Subgroup properties {{group-theory-stub