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...

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...

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...

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...

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...

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 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...

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...

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)...

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...

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...

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...

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...

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...

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

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...

element together with all elements not in any conjugate of H form a normal subgroup called the Frobenius kernel K. (This is a theorem due to Frobenius...

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

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...

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...

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...

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...

or T is normal then the condition ST = TS is satisfied and the product is a subgroup. If both S and T are normal, then the product is normal as well....

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

group of the Cayley graph Γ(G) is isomorphic to the kernel of φ, the normal subgroup of relations among the generators of G. The extreme case is when G...