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Encyclopedia > Normal subgroup

In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has more media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... The word conjugation has several meanings: Grammatical conjugation is the modification of a verb from its basic form. ...

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There are a number of conditions which are equivalent to requiring that a subgroup N be normal in G. Any one of them may be taken as the definition:

  1. For all g in G, g−1NgN.
  2. For all g in G, g−1Ng = N.
  3. The sets of left and right cosets of N in G coincide.
  4. For each g in G, gN = Ng.
  5. N is a union of conjugacy classes of G.

Note that condition (1) is logically weaker than condition (2), and condition (3) is logically weaker than condition (4). For this reason, conditions (1) and (3) are often used to prove that N is normal in G, while conditions (2) and (4) are used to prove consequences of the normality of N in G. In mathematics, if G is a group, H a subgroup of G, and g an element of G, then gH = { gh : h an element of H } is a left coset of H in G, and Hg = { hg : h an element of H } is a right coset of H in G... In set theory and other branches of mathematics, the union of a collection of sets is the set that contains everything that belongs to any of the sets, but nothing else. ... In mathematics, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes reveals many important features of a groups structure. ...


{e} and G are always normal subgroups of G. If these are the only ones, then G is said to be simple. In mathematics, a simple group is a group G such that G is not the trivial group and the only normal subgroups of G are the trivial group and G itself. ...


All subgroups N of an abelian group G are normal, because g−1(Ng) = g−1(gN) = (g−1g)N = N. A group that is not Abelian but for which every subgroup is normal is termed a Hamiltonian group. In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In group theory, a non-abelian group G is called Hamiltonian if every subgroup of G is normal. ...


The normal subgroups of any group G form a lattice under inclusion. The minimum and maximum elements are {e} and G, the greatest lower bound of two normal subgroups is their intersection and their least upper bound is a product group. The term lattice derives from the shape of the Hasse diagrams that result from depicting these orders. ... In mathematics, one can often define a direct product of objects already known, giving a new one. ...


Galois was the first to realize the importance of the existence of normal subgroups. Galois was young-looking for his age and had black hair. ...

Contents


Example

The translation group in any dimension is a normal subgroup of the Euclidean group; for example in 3D: In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ...

  • rotating, translating, and rotating back results in only translation; also reflecting, translating, and reflecting again results in only translation (a translation seen in a mirror looks like a translation, with a reflected translation vector)
  • the left and also the right coset of a rotation by a given angle about a given axis is the set of all rotations by the same angle about a parallel axis, preserving orientation, and those combined with a translation along the axis.
  • the left and also the right coset of a reflection in a plane combined with a rotation by a given angle about a perpendicular axis is the set of all combinations of a reflection in the same or a parallel plane, combined with a rotation by the same angle about the same or a parallel axis, preserving orientation
  • the translations by a given distance in any direction form a conjugacy class; the translation group is the union of those for all distances

Normal subgroups and homomorphisms

Normal subgroups are of relevance because if N is normal, then the quotient group G/N may be formed: if N is normal, we can define a multiplication on cosets by In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ...

(a1N)(a2N) := (a1a2)N

This turns the set of cosets into a group called the quotient group G/N. There is a natural homomorphism f : GG/N given by f(a) = aN. The image f(N) consists only of the identity element of G/N, the coset eN = N. In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ... // Homomorphism for beginners Homomorphism is one of the fundamental concepts in abstract algebra. ...


In general, a group homomorphism f: GH sends subgroups of G to subgroups of H. Also, the preimage of any subgroup of H is a subgroup of G. We call the preimage of the trivial group {e} in H the kernel of the homomorphism and denote it by ker(f). As it turns out, the kernel is always normal and the image f(G) of G is always isomorphic to G/ker(f). In fact, this correspondence is a bijection between the set of all quotient groups G/N of G and the set of all homomorphic images of G (up to isomorphism). It is also easy to see that the kernel of the quotient map, f: GG/N, is N itself, so we have shown that the normal subgroups are precisely the kernels of homomorphisms with domain G. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ... In mathematics, the domain of a function is the set of all input values to the function. ...


Attributes of normality

  • The intersection of a family of normal subgroups is normal
  • The subgroup generated by a family of normal sugroups is normal
  • Normality is preserved upon surjective homomorphisms, and is also preserved upon taking inverse images.
  • Normality is preserved on taking direct products
  • A normal subgroup of a normal subgroup need not be normal. That is, normality is not a transitive property. However, a characteristic subgroup of a normal subgroup is normal. Also, a normal subgroup of a central factor is normal. In particular, a normal subgroup of a direct factor is normal.
  • Every subgroup of index 2 is normal. More generally, a subgroup H of finite index n in G contains a subgroup K normal in G and of index dividing n!.

In abstract algebra, a characteristic subgroup of a group G is a subgroup H of G invariant under each automorphism of G. This means that if f : G → G is a group automorphism (a bijective homomorphism from the group G to itself), then for every x in H we have...

See also


  Results from FactBites:
 
Characteristic subgroup - Wikipedia, the free encyclopedia (468 words)
In mathematics, a characteristic subgroup of a group G is a subgroup H that is invariant under each automorphism of G.
In this case the subgroup H is invariant under the applications of surjective endomorphisms.
The derived subgroup (or commutator subgroup) of a group is always a fully characteristic subgroup, as is the torsion subgroup of an abelian group.
Normal subgroup - Wikipedia, the free encyclopedia (754 words)
In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g
However, a characteristic subgroup of a normal subgroup is normal.
In particular, a normal subgroup of a direct factor is normal.
  More results at FactBites »

 
 

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