In mathematics, especially 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 pronormal 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 ...

that is embedded in a nice way. Pronormality is a simultaneous generalization of both normal subgroups and abnormal subgroups such as

Sylow subgroup In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixe ...

s, . A subgroup is pronormal if each of its conjugates is conjugate to it already in the subgroup generated by it and its conjugate. That is, ''H'' is pronormal in ''G'' if for every ''g'' in ''G'', there is some ''k'' in the subgroup generated by ''H'' and ''H''^''g'' such that ''H''^''k'' = ''H''^''g''. (Here ''H''^''g'' denotes the conjugate subgroup ''gHg''^''-1''.) Here are some relations with other subgroup properties: *Every normal subgroup is pronormal. *Every

is pronormal. *Every pronormal subnormal subgroup is normal. *Every abnormal subgroup is pronormal. *Every pronormal subgroup is weakly pronormal, that is, it has the Frattini property. *Every pronormal subgroup is

paranormal Paranormal events are purported phenomena described in popular culture, folk, and other non-scientific bodies of knowledge, whose existence within these contexts is described as being beyond the scope of normal scientific understanding. Not ...

, and hence polynormal.

References

* Subgroup properties {{Abstract-algebra-stub