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...

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

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...

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...

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...

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...

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...

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...

group theory, a branch of mathematics, a normal p-complement of a finite group for a prime p is a normal subgroup of order coprime to p and index a power...

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...

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

group has a normal subgroup of index p. The focal subgroup theorem relates several lines of investigation in finite group theory: normal subgroups of index...

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)...

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...

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...

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...

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...

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...

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...

groups. A minimal normal subgroup of a group G is a nontrivial normal subgroup N of G such that the only proper subgroup of N that is normal in G is the trivial...

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...

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

subgroup of G then the closure of H is also a subgroup. Likewise, if H is a normal subgroup of G, the closure of H is normal in G. If H is a subgroup...

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...

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...

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

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