In
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 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 ...
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
References
*
*
Subgroup properties
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