Quotient group
In abstract algebra, a quotient group G/N of a group G by a normal subgroup N is the set of all left cosets of N in G, equipped with the group operation defined by (g_1 N)(g_2 N) = (g_1 g_2) N for g_1, g_2 \in G.[1] This construction requires N to be normal in G, meaning that g N g^{-1} = N for all g \in G, or equivalently, that left and right cosets of N coincide, ensuring the operation is well-defined and the resulting structure forms a group.[2] The order of the quotient group G/N equals the index [G : N], the number of distinct cosets, which generalizes Lagrange's theorem by relating subgroup sizes to the structure of G.[2] Quotient groups play a central role in group theory, facilitating the analysis of group homomorphisms through the first isomorphism theorem, which states that for any homomorphism f: G \to H, G / \ker f \cong \operatorname{im} f, where \ker f is the kernel and a normal subgroup of G.[2] Additional isomorphism theorems, such as the second (for subgroups K and normal N, K / (K \cap N) \cong KN / N) and third (for normal chains N \trianglelefteq K \trianglelefteq G, (G/N) / (K/N) \cong G/K), further illustrate how quotients decompose and classify group structures.[2] As a foundational tool in modern algebra, quotient groups enable the study of symmetries and invariants by "factoring out" subgroups, with applications extending to fields like representation theory and classification of finite simple groups.[2] Every subgroup of an abelian group is normal, making quotient constructions particularly straightforward in commutative settings, such as the integers modulo n, which yield cyclic groups essential in number theory.[2]Fundamentals
Definition
A quotient group, also known as a factor group, is constructed from a group G and a normal subgroup N of G.[3] Specifically, G/N denotes the set of all left cosets of N in G, given by \{gN \mid g \in G\}, where each coset gN = \{gn \mid n \in N\} partitions the elements of G.[4] A prerequisite for this construction is that N must be a normal subgroup of G, meaning that for all g \in G and n \in N, the conjugate g^{-1}ng \in N.[4] Normality ensures that the set of cosets forms a group under the induced operation defined by (gN)(hN) = (gh)N for g, h \in G.[3] This operation is well-defined, independent of the choice of representatives g and h, because N is normal.[4] The quotient G/N satisfies the group axioms: the identity element is the coset N itself (corresponding to the identity in G); the inverse of a coset gN is g^{-1}N; and associativity follows from that in G.[3] If G and N are finite, the order of the quotient group is given by |G/N| = |G| / |N|, as established by Lagrange's theorem.[3]Cosets and Normal Subgroups
In group theory, given a group G and a subgroup N \leq G, the left coset of N generated by an element g \in G is the set gN = \{gn \mid n \in N\}.[5] Similarly, the right coset is Ng = \{ng \mid n \in N\}.[5] These cosets represent translates of the subgroup N within G, and in general, left and right cosets may differ when G is non-abelian. The collection of all left cosets of N in G forms a partition of G, meaning the cosets are disjoint and their union is G.[6] The same holds for right cosets.[6] Moreover, every left coset gN and every right coset Ng has the same cardinality as N, so |gN| = |N| and |Ng| = |N| for all g \in G.[6] This equality follows from the bijection h \mapsto gh between N and gN, which preserves the group structure. To endow the set of cosets with a group operation, define multiplication of left cosets by (gN)(hN) = (gh)N. For this operation to be well-defined—independent of the choice of representatives g and h—N must be a normal subgroup of G.[7] Specifically, if g' N = g N and h' N = h N, then g' = g n_1 and h' = h n_2 for some n_1, n_2 \in N, so g' h' = g n_1 h n_2. The product (g' h') N = g (n_1 h) n_2 N equals g h N if and only if n_1 h \in h N, or equivalently, n_1 h = h n_1' for some n_1' \in N. This holds for all such elements precisely when left and right cosets coincide, i.e., when N is normal.[7] A subgroup N \leq G is normal if and only if every left coset equals the corresponding right coset, so gN = Ng for all g \in G.[8] Equivalently, N is normal if it is invariant under conjugation by elements of G, meaning g N g^{-1} = N for all g \in G.[8] These criteria ensure the coset multiplication is associative and forms a group. For an illustration of a non-normal subgroup, consider the symmetric group S_3 on three letters and the subgroup H = \{\mathrm{id}, (1\,3)\}. To check normality, compute the conjugate (1\,2) H (1\,2)^{-1}. Since (1\,2) \cdot \mathrm{id} \cdot (1\,2) = \mathrm{id} and (1\,2) (1\,3) (1\,2) = (2\,3), the conjugate is \{\mathrm{id}, (2\,3)\}, which is not contained in H. Thus, H is not normal in S_3.[9] This failure implies that coset multiplication would not be well-defined for H.Motivation and Construction
Origin of the Term "Quotient"
The term "quotient group" was introduced by William Burnside in his seminal 1897 textbook Theory of Groups of Finite Order, marking the first systematic use of the phrase in the context of abstract group theory.[10] Earlier, in 1893, Arthur Cayley had referred to the structure G/H as a "quotient" without fully specifying the term for the group itself. This naming convention emerged in the late 19th century, paralleling Richard Dedekind's earlier introduction of quotient rings in 1871, where he developed the concept to handle factorization in rings of algebraic integers via ideals.[11] Dedekind's work on quotient structures provided a foundational analogy that influenced group theorists, as both constructions involve dividing an algebraic object by a substructure to form a new entity. The choice of "quotient" reflects a direct analogy to division in arithmetic, particularly integer division. Just as dividing the integers ℤ by the subgroup nℤ yields the cyclic group ℤ/nℤ of order n, the quotient group G/N of a group G by a normal subgroup N has order |G|/|N| when finite, effectively "dividing out" the size of N to obtain a smaller group that captures the structure of G modulo N. This mirrors how remainders in division classify integers into equivalence classes, providing an intuitive bridge from elementary number theory to abstract algebra. Conceptually, forming the quotient group "factors out" the subgroup N by collapsing its elements to the identity, similar to how group presentations mod out by relations to define new groups. This process simplifies the original group by ignoring internal symmetries imposed by N, allowing focus on the coarser structure. In set-theoretic terms, the quotient arises from identifying elements that differ by elements of N, partitioning G into equivalence classes known as cosets, much like quotient sets in general equivalence relations.[12] This identification preserves the group operation on the cosets, yielding a group that encodes G's behavior up to translation by N.Homomorphism Theorem Connection
The first isomorphism theorem establishes a fundamental connection between group homomorphisms and quotient groups. Specifically, if \phi: G \to H is a group homomorphism, then G / \ker(\phi) \cong \operatorname{im}(\phi), where \ker(\phi) denotes the kernel of \phi.[13] If \phi is surjective, this simplifies to G / \ker(\phi) \cong H.[14] A key prerequisite is that the kernel \ker(\phi) must be a normal subgroup of G. To see this, let K = \ker(\phi) and take any g \in G, a \in K. Then \phi(gag^{-1}) = \phi(g) \phi(a) \phi(g)^{-1} = \phi(g) \cdot e \cdot \phi(g)^{-1} = e, where e is the identity in H, so gag^{-1} \in K. Thus, gKg^{-1} \subseteq K for all g \in G, confirming normality.[14] The proof of the theorem proceeds by constructing an induced map from the quotient to the image. Define \psi: G / K \to \operatorname{im}(\phi) by \psi(gK) = \phi(g). This is well-defined because if gK = g'K, then g'^{-1}g \in K, so \phi(g'^{-1}g) = e implies \phi(g') = \phi(g). Moreover, \psi is a homomorphism since \psi((gK)(hK)) = \psi(ghK) = \phi(gh) = \phi(g)\phi(h) = \psi(gK) \psi(hK). It is injective because \psi(gK) = e implies \phi(g) = e, so g \in K and gK = K. Surjectivity follows from the definition of the image. Hence, \psi is an isomorphism.[13] This theorem also explains the construction of quotient groups via projections. For any normal subgroup N \trianglelefteq G, the canonical projection \pi: G \to G/N defined by \pi(g) = gN is a surjective homomorphism with \ker(\pi) = N. Applying the first isomorphism theorem yields G/N \cong \operatorname{im}(\pi) = G/N, which is tautological but confirms the setup.[15] The theorem motivates quotient groups as a universal mechanism for factoring out normal subgroups: any homomorphism vanishing on N factors uniquely through the projection G \to G/N, providing a canonical way to "mod out" by N.[13]Basic Examples
Integers Modulo n
The additive group of integers, denoted \mathbb{Z}, forms an infinite abelian group under addition, with the subgroup n\mathbb{Z} consisting of all integer multiples of a fixed positive integer n. Since \mathbb{Z} is abelian, every subgroup is normal, making n\mathbb{Z} a normal subgroup of \mathbb{Z}.[3] The cosets of n\mathbb{Z} in \mathbb{Z} are the sets of the form k + n\mathbb{Z} for integers k = 0, 1, \dots, n-1, each representing a distinct residue class modulo n. These cosets partition \mathbb{Z} and form the quotient group \mathbb{Z}/n\mathbb{Z}, where the group operation is defined by (a + n\mathbb{Z}) + (b + n\mathbb{Z}) = (a + b) + n\mathbb{Z}. This operation is well-defined because if a' \equiv a \pmod{n} and b' \equiv b \pmod{n}, then a' + b' \equiv a + b \pmod{n}. The quotient group \mathbb{Z}/n\mathbb{Z} is isomorphic to the cyclic group \mathbb{Z}_n of order n, generated by the coset $1 + n\mathbb{Z}.[16] For a concrete illustration, consider n = 6. The cosets are:- $0 + 6\mathbb{Z} = \{\dots, -12, -6, 0, 6, 12, \dots\},
- $1 + 6\mathbb{Z} = \{\dots, -11, -5, 1, 7, 13, \dots\},
- $2 + 6\mathbb{Z} = \{\dots, -10, -4, 2, 8, 14, \dots\},
- $3 + 6\mathbb{Z} = \{\dots, -9, -3, 3, 9, 15, \dots\},
- $4 + 6\mathbb{Z} = \{\dots, -8, -2, 4, 10, 16, \dots\},
- $5 + 6\mathbb{Z} = \{\dots, -7, -1, 5, 11, 17, \dots\}.