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

about c-normal subgroups: Every normal subgroup is c-normal Every retract is c-normal Every c-normal subgroup is weakly c-normal Y. Wang, c-normality...

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...

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...

confused with the normal closure. Clearly C G ( S ) ⊆ N G ( S ) {\displaystyle C_{G}(S)\subseteq N_{G}(S)} and both are subgroups of G {\displaystyle...

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...

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...

{n}},\quad b,c\equiv 0{\pmod {n}}\right\}} This definition immediately implies that Γ ( n ) {\displaystyle \Gamma (n)} is a normal subgroup of finite index...

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...

{\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...

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...

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...

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...

A, B, and C subgroups of a group with A ≤ C (A a subgroup of C) then AB ∩ C = A(B ∩ C); the multiplication here is the product of subgroups. This property...

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...

product of its subgroups G and H. In some contexts, the third property above is replaced by the following: 3′.  Both G and H are normal in P. This property...

_{2}(\mathbb {C} )} , the subgroup of determinant 1 matrices, and N {\displaystyle N} the normal subgroup of scalar matrices C × I = { ( a 0 0 a ) : a ∈ C × } {\displaystyle...

}(G)} to denote the number of maximal and normal subgroups of index n {\displaystyle n} , respectively. Subgroup growth studies these functions, their interplay...

H {\displaystyle f:G\rightarrow H} , let N {\displaystyle N} be a normal subgroup in G {\displaystyle G} and φ {\displaystyle \varphi } the natural surjective...

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...

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...

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

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...

of these normal subgroups, shown with a red background. In this table r means rotations, and f means flips. Because this subgroup is normal, the left...

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

In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order...

subgroup is said to be normal if it is stable under every inner automorphism (which are regular maps). If H {\displaystyle \mathrm {H} } is a normal algebraic...

mathematical field of group theory, Lagrange's theorem states that if H is a subgroup of any finite group G, then | H | {\displaystyle |H|} is a divisor of |...

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