In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, especially in the field of group theory

Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...

, a pronormal subgroup is a subgroup

Subgroup

In group theory, 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 of G if the restriction of * to H x H is a group operation on H...

 that is embedded in a nice way. Pronormality is a simultaneous generalization of both normal subgroup

Normal subgroup

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....

s and abnormal subgroup

Abnormal subgroup

In mathematics, in the field of group theory, an abnormal subgroup is a subgroup H of a group G such that for every x ∈ G, x lies in the subgroup generated by H and H x....

s such as Sylow subgroups, .

A subgroup is pronormal if each of its conjugates

Conjugacy class

In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...

 is conjugate to it already in the subgroup generated

Generating set of a group

In abstract algebra, a generating set of a group is a subset that is not contained in any proper subgroup of the group. Equivalently, a generating set of a group is a subset such that every element of the group can be expressed as the combination of finitely many elements of the subset and their...

 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 are some relations with other subgroup properties:

• Every normal subgroup
  Normal subgroup
  In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
  
   is pronormal.
• Every Sylow subgroup is pronormal.
• Every pronormal subnormal subgroup
  Subnormal subgroup
  In mathematics, in the field of group theory, a subgroup H of a given group G is a subnormal subgroup of G if there is a finite chain of subgroups of the group, each one normal in the next, beginning at H and ending at G....
  
   is normal.
• Every abnormal subgroup
  Abnormal subgroup
  In mathematics, in the field of group theory, an abnormal subgroup is a subgroup H of a group G such that for every x ∈ G, x lies in the subgroup generated by H and H x....
  
   is pronormal.

• Every pronormal subgroup is weakly pronormal, that is, it has the Frattini property
• Every pronormal subgroup is paranormal
  Paranormal subgroup
  In mathematics, in the field of group theory, a paranormal subgroup is a subgroup such that the subgroup generated by it and any conjugate of it, is also generated by it and a conjugate of it within that subgroup....
  
  , and hence polynormal
  Polynormal subgroup
  In mathematics, in the field of group theory, a subgroup of a group 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....