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 ...

, in the field of

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 ( ...

, 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 ...

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 ...

is said to be polynormal if its closure under conjugation by any element of the group can also be achieved via closure by conjugation by some element in the subgroup generated. In symbols, a subgroup

H

of a group

G

is called polynormal if for any

g \in G

the subgroup

K = H^

is the same as

H^

. Here are the relationships with other subgroup properties: * Every weakly

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 ...

is polynormal. * Every paranormal subgroup is polynormal.

References

Subgroup properties {{group-theory-stub