In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation...

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...

{\displaystyle gHg^{-1}} of a subgroup H in G is equal to the index of the normalizer of H in G. If H is a subgroup of G, the index of the normal core of H satisfies...

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. For a group G, the normal core...

maximal subgroups, for example the Prüfer group. Similarly, a normal subgroup N of G is said to be a maximal normal subgroup (or maximal proper normal subgroup)...

series (also normal series, normal tower, subinvariant series, or just series) of a group G is a sequence of subgroups, each a normal subgroup of the next...

form a subgroup of index 2 in S, called the alternating subgroup A. Since A is even a characteristic subgroup of S, it is also a normal subgroup of the...

In group theory, the normal closure of a subset S {\displaystyle S} of a group G {\displaystyle G} is the smallest normal subgroup of G {\displaystyle...

group theory, a subgroup of a group is said to be transitively normal in the group if every normal subgroup of the subgroup is also normal in the whole group...

elements of every subgroup H of G divides the number of elements of G. Cosets of a particular type of subgroup (a normal subgroup) can be used as the...

important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian. In other words, G / N...

a subgroup H, and a normal subgroup N ◃ G {\displaystyle N\triangleleft G} , the following statements are equivalent: G is the product of subgroups, G...

p} . A Sylow p-subgroup (sometimes p-Sylow subgroup) of a finite group G {\displaystyle G} is a maximal p {\displaystyle p} -subgroup of G {\displaystyle...

element is always a normal subgroup of the original group, and the other equivalence classes are precisely the cosets of that normal subgroup. The resulting...

group theory, the Fitting subgroup F of a finite group G, named after Hans Fitting, is the unique largest normal nilpotent subgroup of G. Intuitively, it...

monomorphism f from H to G is normal if and only if its image is a normal subgroup of G. In particular, if H is a subgroup of G, then the inclusion map...

group G: G has a central series of finite length. That is, a series of normal subgroups { 1 } = G 0 ◃ G 1 ◃ ⋯ ◃ G n = G {\displaystyle \{1\}=G_{0}\triangleleft...

product S N {\displaystyle SN} is a subgroup of G {\displaystyle G} , The subgroup N {\displaystyle N} is a normal subgroup of S N {\displaystyle SN} , The...

of the cyclic groups. Z 4 {\displaystyle \mathbb {Z} _{4}} is not a normal subgroup. A group G is called solvable if it has a subnormal series whose factor...

characteristic subgroup is normal; though the converse is not guaranteed. Examples of characteristic subgroups include the commutator subgroup and the center...

In mathematics, specifically group theory, a Hall subgroup of a finite group G is a subgroup whose order is coprime to its index. They were introduced...

connected normal solvable subgroup Gnil for the largest connected normal nilpotent subgroup so that we have a sequence of normal subgroups 1 ⊆ Gnil ⊆...

describes the relation between representations of a group and those of a normal subgroup. Alfred H. Clifford proved the following result on the restriction...

Discrete normal subgroups play an important role in the theory of covering groups and locally isomorphic groups. A discrete normal subgroup of a connected...

term in the series is a normal subgroup of its successor. The series may be infinite. If the series is finite, then the subgroup is subnormal. automorphism...

field of group theory, a subgroup H {\displaystyle H} of a group G {\displaystyle G} is called c-normal if there is a normal subgroup T {\displaystyle T} of...

connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted SO(n). It consists...

contains normal subgroups of order pi with 0 ≤ i ≤ n, and any normal subgroup of order pi is contained in the ith center Zi. If a normal subgroup is not...

group whose commutator subgroup is abelian. Equivalently, a group G is metabelian if and only if there is an abelian normal subgroup A such that the quotient...

In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple...