Basic notions in group theory
category of groups
subgroups, normal subgroups
quotient groups
group homomorphisms, kernel, image
(semi-)direct product, direct sum
types of groups
finite, infinite
discrete, continuous
multiplicative, additive
abelian, cyclic, simple, solvable

In mathematics, more specifically in abstract algebra, a normal subgroup is a special kind of subgroup. In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Mathematics, the word kernel has several meanings Kernel may mean a subset associated with a mapping The kernel of a mapping is the set of elements that In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In Mathematics, one can often define a direct product of objectsalready known giving a new one The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, a finite group is a group which has finitely many elements Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a discrete group is a group G equipped with the Discrete topology. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak In Mathematics, an additive group may be an Abelian group, when it is written using the symbol + for its Binary operation An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of Normal subgroups are important because they can be used to construct quotient groups from a given group. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element

Évariste Galois was the first to realize the importance of the existence of normal subgroups.

Contents 1 Definitions 2 Examples 3 Properties 3.1 Lattice of normal subgroups 4 Normal subgroups and homomorphisms 5 See also 6 References 7 External links

Definitions

A subgroup N of a group G is called a normal subgroup if it is invariant under conjugation; that is, for each element n in N and each g in G, the element gng⁻¹ is still in N. In a group, the conjugate by g of h is ghg -1 Translation If h is a translation then its conjugate by We write

The following conditions are equivalent to requiring that a subgroup N be normal in G. In Logic, statements p and q are logically equivalent if they have the same logical content Any one of them may be taken as the definition:

1. For all g in G, gNg⁻¹ ⊆ N.
2. For all g in G, gNg⁻¹ = N.
3. The sets of left and right cosets of N in G coincide. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH
4. For all g in G, gN = Ng.
5. N is a union of conjugacy classes of G. In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets In Mathematics, especially Group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class
6. There is some homomorphism on G for which N is the kernel. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism

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.

Examples

• {e} and G are always normal subgroups of G. These groups are called the trivial subgroups, and if these are the only ones, then G is said to be simple. SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning

• The center of a group is a normal subgroup. In Abstract algebra, the center of a group G is the set Z ( G) of all elements in G which commute with all the

• The commutator subgroup is a normal subgroup. In Mathematics, more specifically in Abstract algebra, the commutator subgroup or derived subgroup of a group is the Subgroup

• More generally, any characteristic subgroup is normal, since conjugation is always an automorphism. In Mathematics, a characteristic subgroup of a group G is a Subgroup H that is invariant under each Automorphism of In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself

• All subgroups N of an abelian group G are normal, because gN = Ng. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the A group that is not abelian but for which every subgroup is normal is called a Hamiltonian group. In Group theory, a Dedekind group is a group G such that every Subgroup of G is normal.

• The translation group in any dimension is a normal subgroup of the Euclidean group; for example in 3D 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). In Euclidean geometry, a translation is moving every point a constant distance in a specified direction In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional The translations by a given distance in any direction form a conjugacy class; the translation group is the union of those for all distances.

• In the Rubik's Cube group, the subgroup consisting of operations which only affect the corner pieces is normal, because no conjugate transformation can make such an operation affect an edge piece instead of a corner. The Rubik's Cube provides a tangible representation of a mathematical group. By contrast, the subgroup consisting of turns of the top face only is not normal, because a conjugate transformation can move parts of the top face to the bottom and hence not all conjugates of elements of this subgroup are contained in the subgroup.

Properties

• 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 of a group need not be normal in the group. That is, normality is not a transitive relation. In Mathematics, a Binary relation R over a set X is transitive if whenever an element a is related to an element b However, a characteristic subgroup of a normal subgroup is normal. In Mathematics, a characteristic subgroup of a group G is a Subgroup H that is invariant under each Automorphism of 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. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH More generally, a subgroup H of finite index n in G contains a subgroup K normal in G and of index dividing n! called the normal core. In Group theory, a branch of Mathematics, the term core is used to denote special Normal subgroups of a group. In particular, if p is the smallest prime dividing the order of G, then every subgroup of index p is normal.

Lattice of normal subgroups

The normal subgroups of a group G form a lattice under subset inclusion with least element {e} and greatest element G. In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' In Mathematics, especially in Order theory, the greatest element of a subset S of a Partially ordered set (poset is an element of S In Mathematics, especially in Order theory, the greatest element of a subset S of a Partially ordered set (poset is an element of S Given two normal subgroups N and M in G, meet is defined as

and join is defined as

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

(a₁N)(a₂N) := (a₁a₂)N. In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' In Mathematics, a lattice is a Partially ordered set (also called a poset) in which every pair of elements has a unique Supremum (the elements' In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G

This turns the set of cosets into a group called the quotient group G/N. There is a natural homomorphism f : G → G/N given by f(a) = aN. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector The image f(N) consists only of the identity element of G/N, the coset eN = N.

In general, a group homomorphism f: G → H 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). In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism As it turns out, the kernel is always normal and the image f(G) of G is always isomorphic to G/ker(f) (the first isomorphism theorem). In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural 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). 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 It is also easy to see that the kernel of the quotient map, f: G → G/N, is N itself, so we have shown that the normal subgroups are precisely the kernels of homomorphisms with domain G. In Mathematics, the domain of a given function is the set of " Input " values for which the function is defined

See also

References

• I. N. Herstein, Topics in algebra. In Group theory, the centralizer and normalizer of a Subset S of a group G are Subgroups of G which In Group theory, the conjugate closure of a Subset S of a group G is the Subgroup of G generated In Group theory, a branch of Mathematics, the term core is used to denote special Normal subgroups of a group. In Mathematics, a characteristic subgroup of a group G is a Subgroup H that is invariant under each Automorphism of In Mathematics, a Subgroup of a group is fully characteristic (or fully invariant) if it is Invariant under every Endomorphism In Mathematics, in the field of Group theory, a Subgroup H of a given group G is a subnormal subgroup of G if In Mathematics, in the field of Group theory, a Subgroup of a group is said to be ascendant if there is an ascending series starting from the In Mathematics, in the field of Group theory, a quasinormal subgroup, or permutable subgroup, is a Subgroup of a group that commutes In Mathematics, in the field of Group theory, a Subgroup A of a group G is termed seminormal if there is a subgroup B In Mathematics, in the field of Group theory, a conjugate permutable subgroup is a Subgroup that commutes with all its conjugate subgroups In Mathematics, in the field of Group theory, a modular subgroup is a Subgroup that is a modular element in the Lattice of subgroups In Mathematics, especially in the field of Group theory, a pronormal subgroup is a Subgroup that is embedded in a nice way In Mathematics, in the field of Group theory, a paranormal subgroup is a Subgroup such that the subgroup generated by it and any conjugate In Mathematics, in the field of Group theory, a Subgroup of a group is said to be polynormal if its closure under conjugation by any In Mathematics, in the field of Group theory, a Subgroup H of a group G is called c normal if there is a Normal In Mathematics, in the field of Group theory, a Subgroup H of a group G is termed malnormal if for any x In Mathematics, in the field of Group theory, a contranormal subgroup is a Subgroup whose Normal closure in the group is the whole group In Mathematics, in the field of Group theory, an abnormal subgroup is a Subgroup H of a group G such that for any x In Group theory, the centralizer and normalizer of a Subset S of a group G are Subgroups of G which In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. Israel Nathan Herstein ( March 28, 1923, Lublin, Poland – February 9, 1988, Chicago, Illinois) was Second edition. Xerox College Publishing, Lexington, Mass. -Toronto, Ont. , 1975. xi+388 pp.
• David S. Dummit; Richard M. Foote, Abstract algebra. Prentice Hall, Inc. , Englewood Cliffs, NJ, 1991. xiv+658 pp. ISBN 0-13-004771-6

External links

• Eric W. Weisstein, normal subgroup at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
• Normal subgroup in Springer's Encyclopedia of Mathematics
• Robert Ash: Group Fundamentals in Abstract Algebra. The Basic Graduate Year

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