In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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, 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 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 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
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...
of its successor.
The series may be infinite. If the series is finite, then the subgroup is
subnormal. 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.
See also
*
Descendant subgroup
In mathematics, in the field of group theory, a subgroup of a group is said to be descendant if there is a descending series starting from the subgroup and ending at the group, such that every term in the series is a normal subgroup of its predec ...
References
*
*
Subgroup properties
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