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...

, 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 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...

 of a group

Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...

 is said to be ascendant if there is an ascending series starting from the subgroup and ending at the group, such that every term in the series is a 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....

 of its successor.

The series may be infinite. If the series is finite, then the subgroup is subnormal

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....

. Here are some properties of ascendant subgroups:

• Every subnormal subgroup is ascendant; every ascendant subgroup is serial.
• In a finite group, the properties of being ascendant and subnormal are equivalent.
• An arbitrary intersection of ascendant subgroups is ascendant.
• Given any subgroup, there is a minimal ascendant subgroup containing it.
• Every subgroup can be expressed uniquely as an ascendant subgroup of a self-normalizing subgroup.