Normal subgroup information

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

as "H is a subgroup of G". The trivial subgroup of any group is the subgroup {e} consisting of just the identity element. A proper subgroup of a group...

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...

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...

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

{\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...

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 p} -subgroup of G {\displaystyle G} is a normal subgroup. However, there are groups that have normal subgroups but no normal Sylow subgroups, such as...

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...

e, a subgroup H, and a normal subgroup N ◁ G, the following statements are equivalent: G is the product of subgroups, G = NH, and these subgroups have...

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 subgroup invariant under conjugation Normal (album), a 2005 album by Ron "Bumblefoot" Thal "Normal" (Alonzo song) "Normal" (Eminem song) "Normal", a...

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...

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...

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

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...

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)...

the identity element is always a normal subgroup, and the other equivalence classes are the other cosets of this subgroup. Together, these equivalence classes...

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...

is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q {\displaystyle Q} and N {\displaystyle N}...

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...

of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the general linear group given by the kernel of the determinant...

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...

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

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 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...