In the
mathematical
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 ...
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 ...
''H'' of a given
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 ide ...
''G'' is a serial subgroup of ''G'' if there is a chain ''C'' of subgroups of ''G'' extending from ''H'' to ''G'' such that for consecutive subgroups ''X'' and ''Y'' in ''C'', ''X'' 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 ''Y''.
The relation is written ''H ser G'' or ''H is serial in G''.
If the chain is finite between ''H'' and ''G'', then ''H'' is a
subnormal subgroup of ''G''. Then every subnormal subgroup of ''G'' is serial. If the chain ''C'' is well-ordered and ascending, then ''H'' is an
ascendant subgroup
In mathematics, in the field of group theory, 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 id ...
of ''G''; if descending, then ''H'' is a
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 ...
of ''G''. If ''G'' is a
locally finite group In mathematics, in the field of group theory, a locally finite group is a type of group that can be studied in ways analogous to a finite group. Sylow subgroups, Carter subgroups, and abelian subgroups of locally finite groups have been studied. T ...
, then the set of all serial subgroups of ''G'' form a
complete sublattice in the
lattice
Lattice may refer to:
Arts and design
* Latticework, an ornamental criss-crossed framework, an arrangement of crossing laths or other thin strips of material
* Lattice (music), an organized grid model of pitch ratios
* Lattice (pastry), an orna ...
of all normal subgroups of ''G''.
See also
*
Characteristic subgroup
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphism ...
*
Normal closure
*
Normal core In group theory, a branch of mathematics, a core is any of certain special normal subgroups of a group. The two most common types are the normal core of a subgroup and the ''p''-core of a group.
The normal core Definition
For a group ''G'', the n ...
References
Subgroup properties
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