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Coset

In group theory, a coset of a subgroup H in a group G is a subset of G formed by multiplying (or adding, in additive notation) every element of H by a fixed element g \in G, resulting in either a left coset gH = \{gh \mid h \in H\} or a right coset Hg = \{hg \mid h \in H\}.^[1]^[2] These cosets partition the group G into disjoint subsets of equal cardinality to H, with each element of G belonging to exactly one coset of H.^[3]^[4] The concept of cosets is fundamental to understanding the structure of groups, as it enables the classification of elements relative to subgroups and leads to key results like Lagrange's theorem, which states that the order of H divides the order of G if G is finite, with the number of distinct cosets (the index [G:H]) equaling |G|/|H|.^[5]^[6] In abelian groups, left and right cosets coincide, simplifying the analysis, but in non-abelian groups, they may differ, highlighting asymmetries in the group operation.^[2] Cosets also form the basis for quotient groups G/H, where the cosets serve as elements under a well-defined operation, provided H is normal (i.e., left and right cosets match for all g \in G).^[4]^[7] Beyond finite groups, cosets apply to infinite groups and have applications in symmetry studies, coding theory, and algebraic structures like rings and modules, where analogous notions (e.g., ideals) play similar roles.^[5] The index of a subgroup, determined by the number of cosets, quantifies how H "sits inside" G and is crucial for theorems on solvability and representation theory.^[6]

Fundamentals

Definition

In group theory, a subgroup H of a group G is a nonempty subset of G that is closed under the group operation, contains the identity element, and is closed under taking inverses, thereby forming a group under the restriction of G's operation.^[8] Given a group G and a subgroup H \leq G, the left coset of H containing an element g \in G is the set gH = \{ gh \mid h \in H \}.^[9]^[10] Similarly, the right coset of H containing g is the set Hg = \{ hg \mid h \in H \}.^[9]^[10] The standard notation uses gH for left cosets and Hg for right cosets to distinguish the side on which the subgroup elements act.^[9] In abelian groups, where the operation is commutative, left and right cosets coincide since gh = hg for all g \in G and h \in H.^[9]

Basic Example

A concrete example of cosets arises in the symmetric group S_3, which consists of all permutations of the set \{1, 2, 3\} and has six elements: the identity e, the transpositions (12), (13), (23), and the 3-cycles (123), (132).^[5] Consider the subgroup H = \{e, (12)\}, which is the stabilizer of the point 3 under the action of S_3 on \{1, 2, 3\}.^[5] The left cosets of H in S_3 are computed by left multiplication:

eH = H = \{e, (12)\},
(13)H = \{(13), (123)\},
(23)H = \{(23), (132)\}.

These three left cosets partition S_3, as each has two elements and |S_3|/|H| = 6/2 = 3.^[5] The right cosets of H in S_3 are:

He = H = \{e, (12)\},
H(13) = \{(13), (132)\},
H(23) = \{(23), (123)\}.

Again, these partition S_3 into three cosets of order two.^[5] Comparing the cosets reveals that the left cosets differ from the right cosets—for instance, (13)H = \{(13), (123)\} while H(13) = \{(13), (132)\}—indicating that H is not a normal subgroup of S_3.^[5] To visualize the partitioning: Left Cosets:

\{e, (12)\}
\{(13), (123)\}
\{(23), (132)\}

Right Cosets:

\{e, (12)\}
\{(13), (132)\}
\{(23), (123)\}^[5]

Properties

Partition and Equivalence

In group theory, the left cosets of a subgroup H of a group G form a partition of G. This means that the distinct left cosets are pairwise disjoint, and their union equals G. Specifically, for any g, g' \in G, if gH \cap g'H \neq \emptyset, then gH = g'H, ensuring disjointness for unequal cosets; moreover, every element x \in G belongs to the coset xH, guaranteeing completeness of the partition.^[11] This partitioning arises from an equivalence relation \sim_H on G defined by g \sim_H g' if and only if g^{-1}g' \in H. The relation \sim_H is reflexive because g^{-1}g = e \in H for all g \in G; symmetric since if g^{-1}g' \in H, then (g')^{-1}g = (g^{-1}g')^{-1} \in H; and transitive because if g^{-1}g' \in H and (g')^{-1}g'' \in H, then g^{-1}g'' = (g^{-1}g')(g')^{-1}g'' \in H. The equivalence class of g under \sim_H is precisely the left coset gH.^[12] To see disjointness more directly, suppose gH \cap g'H \neq \emptyset, so there exist h, h' \in H with gh = g'h'; then g^{-1}g' = h(h')^{-1} \in H, implying g' \in gH and thus g'H \subseteq gH, and symmetrically gH \subseteq g'H, so gH = g'H. Completeness follows as every g \in G lies in gH.^[11] All left cosets of H have the same cardinality as H. The map h \mapsto gh defines a bijection from H to gH, as it is injective by left cancellation in G (if gh_1 = gh_2, then h_1 = h_2) and surjective by definition of the coset. The same holds for right cosets.^[5]

Index and Cardinality

The index of a subgroup H in a group G, denoted [G : H], is defined as the number of distinct left cosets of H in G; this coincides with the number of right cosets since the cosets partition G regardless of the side chosen.^[5]^[12] For finite groups, the index satisfies the formula [G : H] = |G| / |H|, where |G| and |H| denote the orders of G and H, respectively.^[5]^[12] This relation is a direct consequence of Lagrange's theorem, which states that if G is a finite group and H is a subgroup, then |H| divides |G|, or equivalently, |G| = [G : H] \cdot |H|.^[5]^[12] The proof relies on the partition of G into [G : H] cosets, each of which has the same cardinality as H, so the total order of G is the product of the index and the subgroup order; a bijection between elements of a coset gH and H via left multiplication by g establishes the equal sizes.^[12] In infinite groups, the index [G : H] can be finite, countably infinite, or uncountably infinite, depending on the number of distinct cosets.^[5] For example, in the additive group of integers \mathbb{Z} with the subgroup $2\mathbb{Z} of even integers, the left cosets are $2\mathbb{Z} (evens) and $1 + 2\mathbb{Z} (odds), yielding index [\mathbb{Z} : 2\mathbb{Z}] = 2.^[5]

Normal Subgroups

A subgroup N of a group G is called normal, denoted N \trianglelefteq G, if the left coset gN equals the right coset Ng for every g \in G.^[13] Equivalently, N is normal if gNg^{-1} = N for all g \in G, meaning N is invariant under conjugation by elements of G.^[14] The condition that left and right cosets coincide for every g \in G is precisely what characterizes normality: a subgroup N is normal if and only if its left and right cosets are the same./05%3A_Cosets_Lagranges_Theorem_and_Normal_Subgroups/5.03%3A_Normal_Subgroups) If N \trianglelefteq G, the set of all cosets of N in G forms a group called the quotient group G/N, with the group operation defined by (gN)(g'N) = (gg')N for g, g' \in G.^[13] This operation is well-defined precisely because N is normal, ensuring that the product of cosets depends only on the cosets themselves and not on the choice of representatives. The identity element is N itself, and the inverse of gN is g^{-1}N. For example, in the symmetric group S_3, the alternating group A_3 = \langle (123) \rangle is normal because its left and right cosets coincide: the cosets are A_3 and (12)A_3 = A_3(12).^[13] The trivial subgroup \{e\} is always normal in any group G, as g\{e\} = \{e\}g = \{e\} for all g \in G, and similarly the whole group G is normal in itself./05%3A_Cosets_Lagranges_Theorem_and_Normal_Subgroups/5.03%3A_Normal_Subgroups)

Examples

Additive Groups of Integers

In the additive group of integers \mathbb{Z} under addition, a fundamental example of a subgroup is n\mathbb{Z}, consisting of all integer multiples of a fixed positive integer n. This subgroup includes elements such as \dots, -2n, -n, 0, n, 2n, \dots and is itself an infinite cyclic group generated by n.^[5] The cosets of n\mathbb{Z} in \mathbb{Z} are sets of the form k + n\mathbb{Z} = \{k + nm \mid m \in \mathbb{Z}\} for any integer k. These cosets represent equivalence classes where two integers are equivalent if their difference is a multiple of n, corresponding directly to residue classes modulo n. The distinct cosets are precisely those with representatives k = 0, 1, \dots, n-1, as any integer a satisfies a \equiv r \pmod{n} for some r in this range, placing a in the coset r + n\mathbb{Z}.^[5]^[15] Since \mathbb{Z} is an abelian group, left cosets and right cosets coincide for any subgroup: k + n\mathbb{Z} = n\mathbb{Z} + k. The index [\mathbb{Z} : n\mathbb{Z}], or the number of distinct cosets, equals n, and these n cosets partition \mathbb{Z} into disjoint subsets whose union is the entire group.^[5]^[15] The collection of these cosets forms the quotient group \mathbb{Z}/n\mathbb{Z}, known as the integers modulo n, where addition is defined by (k + n\mathbb{Z}) + (l + n\mathbb{Z}) = (k + l) + n\mathbb{Z}, equivalent to addition modulo n. This structure is a cyclic group of order n, illustrating how cosets generalize modular arithmetic in group theory.^[5]^[15]

Vector Spaces

In a vector space V over a field F, given a subspace W \subseteq V, the coset of W containing a vector v \in V is the set v + W = \{v + w \mid w \in W\}, which forms an affine subspace of V.^[16]^[17] Under the additive group structure of V, the cosets of W partition V into disjoint affine subspaces, with the index [V : W] (the number of distinct cosets) being infinite unless V is finite-dimensional over a finite field.^[16]^[18] For example, in the vector space \mathbb{R}^2 with subspace W as the x-axis \{(x, 0) \mid x \in \mathbb{R}\}, each coset (0, c) + W = \{(x, c) \mid x \in \mathbb{R}\} is a horizontal line y = c parallel to the x-axis.^[5]^[17] All such cosets are parallel to W and arise as translates of W by elements of the quotient space V/W, which inherits a vector space structure from V.^[16]^[19] The dimension of the quotient space satisfies \dim(V/W) = \dim V - \dim W, measuring the "codimension" of W in V.^[19]^[20]

Matrix Groups

In matrix groups such as the general linear group \mathrm{GL}(n, \mathbb{R}), cosets provide a way to understand the structure of subgroups like the unipotent upper triangular matrices. Consider the subgroup U of \mathrm{GL}(2, \mathbb{R}) consisting of matrices of the form

\begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix},

where a \in \mathbb{R}. This subgroup is unipotent, with all elements having eigenvalues equal to 1, and it forms the unipotent radical of the Borel subgroup of upper triangular matrices in \mathrm{GL}(2, \mathbb{R}).^[21] The left coset of an arbitrary invertible matrix A = \begin{pmatrix} p & q \\ r & s \end{pmatrix} (with ps - qr \neq 0) by U is A U = \left\{ \begin{pmatrix} p & q + p b \\ r & s + r b \end{pmatrix} \mid b \in \mathbb{R} \right\}. Matrices in this coset have fixed first column (p, r) and second column obtained by adding b times the first column to (q, s). This reflects how left multiplication by elements of U adds multiples of the first column to the second column.^[21] In contrast, the right coset U A = \left\{ \begin{pmatrix} p + b r & q + b s \\ r & s \end{pmatrix} \mid b \in \mathbb{R} \right\} fixes the second row (r, s) and varies the first row (p, q) by adding arbitrary multiples of the second row. This difference between left and right cosets illustrates that U is not normal in \mathrm{GL}(2, \mathbb{R}); for instance, conjugation by the Weyl element w = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} maps U to the opposite unipotent subgroup of lower triangular matrices with 1s on the diagonal.^[21] The index of U in \mathrm{GL}(2, \mathbb{R}) is infinite, as both groups have the cardinality of the continuum, and the cosets partition \mathrm{GL}(2, \mathbb{R}) into uncountably many classes corresponding to choices of basis vectors modulo the action of U. In finite field analogs, such as \mathrm{GL}(2, \mathbb{F}_q), the corresponding unipotent subgroup has finite index (q-1)^2 (q+1), and finite subgroups like the scalar matrices \{ \lambda I \mid \lambda \in \mathbb{F}_q^\times \} also have finite index q(q-1)(q+1).^[21] In linear representations, cosets of U relate to stabilizers of flags; specifically, U stabilizes the standard flag \mathbb{R} e_2 \subset \mathbb{R}^2 under the natural action of \mathrm{GL}(2, \mathbb{R}) on \mathbb{R}^2, making left cosets useful for classifying orbits of such flags in representation theory.^[21]

Advanced Concepts

Cosets as Orbits

In group theory, cosets can be interpreted through the lens of group actions and their associated orbits, providing a deeper connection to the structure of groups and subgroups. Consider a group G acting on itself by left multiplication, defined by the map g \cdot x = gx for g, x \in G. This defines a transitive action, where the orbit of any element x \in G is \mathrm{Orb}(x) = Gx = \{gx \mid g \in G\} = G, the entire group, since every element can be reached from x by appropriate multiplication.^[22] The stabilizer of x under this action is \mathrm{Stab}_G(x) = \{g \in G \mid gx = x\} = \{e\}, the trivial subgroup, reflecting the faithfulness of the action.^[23] For a subgroup H \leq G, a related perspective arises from the conjugation action of G on the set of all subgroups, but more directly, cosets emerge as orbits under actions induced by H itself. Specifically, H acts on G by right multiplication, a right group action given by g \cdot h = gh for g \in G and h \in H. The orbit of an element g \in G under this action is \mathrm{Orb}_H(g) = \{gh \mid h \in H\} = gH, which is precisely the left coset of H containing g. Thus, the left cosets of H in G partition G into the orbits of this action, with each orbit having cardinality |H| since the stabilizer of g is trivial: \{h \in H \mid gh = g\} = \{e\}.^[24] This view underscores that the cosets form an equivariant decomposition of G under the subgroup's influence.^[22] The orbit-stabilizer theorem further illuminates this structure. For a group G acting on a set X, the theorem states that for any x \in X, |G| = |\mathrm{Orb}(x)| \cdot |\mathrm{Stab}_G(x)|, or more generally, the orbit is in bijection with the set of left cosets G / \mathrm{Stab}_G(x).^[23] In the context of cosets, consider G acting on the set of left cosets G/H = \{gH \mid g \in G\} by left multiplication: k \cdot (gH) = (kg)H. This action is transitive, with a single orbit equal to G/H. The stabilizer of the coset H (the "identity" coset) is \mathrm{Stab}_G(H) = \{k \in G \mid kH = H\} = H. Applying the orbit-stabilizer theorem yields |G/H| = [G : H] = |G| / |H|, confirming the index as the orbit size relative to the stabilizer.^[25] If H is normal, the orbits remain invariant under conjugation, aligning with the quotient group structure.^[22] Dually, right cosets arise from the left action of H on G defined by h \cdot g = h g for h \in H, g \in G. The orbit of g under this action is \mathrm{Orb}_H(g) = \{ h g \mid h \in H \} = H g, which is the right coset containing g. The stabilizer of g is \{ h \in H \mid h g = g \} = \{ e \}, and analogous stabilizer and index relations hold by symmetry.^[26] This duality highlights how left and right cosets correspond to opposite-sided actions, each partitioning G equivariantly.^[23]

Double Cosets

In group theory, given a group G and subgroups H, K \leq G, the double coset of an element g \in G with respect to H and K is the set HgK = \{ h g k \mid h \in H, k \in K \}.^[27] This generalizes the notion of a single coset by incorporating elements from two distinct subgroups on either side of g.^[5] The distinct double cosets HgK form a partition of G, meaning they are pairwise disjoint (or equal) and their union equals G.^[27] This partitioning property mirrors that of left or right cosets for a single subgroup, but double cosets generally vary in size and structure depending on the choice of representative g.^[5] The cardinality of a double coset is given by the formula |HgK| = \frac{|H| \cdot |K|}{|H \cap gKg^{-1}|}, which reflects the overlap between H and the conjugate gKg^{-1}.^[27] For a concrete illustration, consider the symmetric group S_3 with H = \langle (1\,2) \rangle = \{ e, (1\,2) \} and K = \langle (1\,3) \rangle = \{ e, (1\,3) \}. The double cosets are HK = \{ e, (1\,2), (1\,3), (1\,3\,2) \} of size 4, and H(2\,3)K = \{ (2\,3), (1\,2\,3) \} of size 2.^[5] These examples demonstrate how double cosets can have sizes that do not divide |G| = 6, unlike the sizes of single cosets.^[5] Double cosets are essential in advanced topics, such as Frobenius reciprocity, where they index the decomposition of induced representations between subgroups.^[28]

Applications

Coding Theory

In coding theory, particularly for linear error-correcting codes, cosets of a code subspace partition the ambient vector space into equivalence classes that facilitate efficient decoding. A linear code C over the finite field \mathbb{F}_q is defined as a k-dimensional subspace of the n-dimensional vector space V = \mathbb{F}_q^n, where n is the code length and q is the field size. The cosets of C in V are the sets C + \mathbf{u} = \{\mathbf{c} + \mathbf{u} \mid \mathbf{c} \in C\} for \mathbf{u} \in V, and these cosets form a partition of V into q^{n-k} distinct classes, each corresponding to a unique equivalence relation where \mathbf{v} \sim \mathbf{w} if \mathbf{v} - \mathbf{w} \in C. This structure arises from the quotient space V/C, enabling the analysis of error patterns in transmitted codewords.^[29] Within each coset, the coset leader is the vector of minimum Hamming weight, serving as the representative error pattern assumed during decoding to minimize the likelihood of incorrect correction. The Hamming weight of a vector counts the number of nonzero entries, and selecting the coset leader ensures that decoding corrects errors up to the code's designed capability, typically half the minimum distance d of C. Coset leaders are precomputed for practical codes, often stored in decoding tables that map coset representatives to their leaders.^[30] Syndrome decoding leverages cosets by using a parity-check matrix H of C, an (n-k) \times n matrix whose rows span the dual code C^\perp, satisfying H\mathbf{c} = \mathbf{0} for all \mathbf{c} \in C. For a received vector \mathbf{r} = \mathbf{c} + \mathbf{e} (where \mathbf{e} is the error vector), the syndrome \mathbf{s} = H\mathbf{r} = H\mathbf{e} is computed, as H\mathbf{c} = \mathbf{0}; this \mathbf{s} \in \mathbb{F}_q^{n-k} uniquely identifies the coset C + \mathbf{r}, since the kernel of H is exactly C. The number of possible syndromes equals the number of cosets, q^{n-k}, allowing syndromes to act as compact labels for coset lookup.^[31] The standard syndrome decoding algorithm proceeds as follows: compute the syndrome \mathbf{s} of the received \mathbf{r}; retrieve the coset leader \mathbf{e}' associated with \mathbf{s} from a prebuilt table or lookup mechanism; and estimate the transmitted codeword as \hat{\mathbf{c}} = \mathbf{r} - \mathbf{e}'. This corrects all error patterns \mathbf{e} where the weight of \mathbf{e} - \mathbf{e}' exceeds that of \mathbf{e}' only if the table is designed for the code's error-correcting radius t = \lfloor (d-1)/2 \rfloor. The approach is efficient for codes where syndrome computation and table access are feasible, reducing the search space from q^n to q^{n-k}.^[29] A canonical example is the binary Hamming code, a [7,4,3] linear code over \mathbb{F}_2 that corrects single errors using coset-based syndrome decoding. Its parity-check matrix H is a $3 \times 7 matrix with columns consisting of all distinct nonzero binary vectors of length 3, ensuring minimum distance 3. The 8 possible syndromes (including zero) correspond to the 8 cosets: the zero syndrome indicates no error (coset leader \mathbf{0}), while each of the 7 nonzero syndromes matches a column of H, identifying the position of a single-bit error as the coset leader of weight 1. Decoding subtracts this leader from \mathbf{r} to recover \mathbf{c}.^[32] This Hamming code is perfect, meaning its cosets' spheres of radius 1 (each containing 1 + 7 = 8 vectors) exactly cover \mathbb{F}_2^7 without overlap, so every received vector lies in a unique coset with leader weight at most 1, enabling reliable single-error correction for all patterns of weight up to 1.^[33]

Representation Theory

In representation theory, cosets play a fundamental role in constructing induced representations from a subgroup to the full group. Given a finite group G, a subgroup H \leq G, and a representation \rho: H \to \mathrm{[GL](/page/GL)}(V) of H on a vector space V, the induced representation \mathrm{[Ind](/page/IND)}_H^G(\rho) is defined on the vector space \bigoplus_{gH \in G/H} V_{gH}, where each V_{gH} is a copy of V. The action of G permutes the coset components: for x \in G, the operator \mathrm{[Ind](/page/IND)}_H^G(\rho)(x) maps v \in V_{gH} to \rho(h^{-1})(v) \in V_{g'hH} if gxH = g'hH with h \in H, effectively extending \rho across the cosets via conjugation and permutation.^[34] This construction relies on the left cosets G/H to decompose the representation space, ensuring the induced module captures the action of G on functions or sections equivariant under H.^[35] A key tool involving double cosets is the Mackey formula, which decomposes the restriction of an induced representation to another subgroup. For subgroups H, K \leq G and an H-representation \rho, the restriction \mathrm{Res}_K^G \left( \mathrm{Ind}_H^G(\rho) \right) decomposes as \bigoplus_{s \in H \backslash G / K} \mathrm{Ind}_{K \cap sHs^{-1}}^K \left( \rho^s |_{K \cap sHs^{-1}} \right), where \rho^s(y) = \rho(s y s^{-1}) for y \in K \cap s H s^{-1}, and the sum runs over a set of double coset representatives H s K.^[36] Double cosets H s K = \{ h s k \mid h \in H, k \in K \} partition G and index the summands, providing a way to break down the representation into induced pieces from the intersections stabilized by conjugation. This formula is essential for analyzing tensor products like \mathrm{Ind}_H^G(\rho) \otimes \mathrm{Ind}_K^L(\sigma) via double coset decompositions.^[36] For finite groups, the character of the induced representation admits an explicit formula in terms of coset transversals. If \chi is the character of \rho, and X is a transversal for the left cosets G/H, then the character \chi^G of \mathrm{Ind}_H^G(\rho) is given by

\chi^G(g) = \frac{1}{|H|} \sum_{x \in X} \chi(x^{-1} g x),

where \chi(x^{-1} g x) = 0 if x^{-1} g x \notin H.^[34] This sums the original character values over conjugates landing in H, weighted by the index [G:H] = |X|, and directly uses the coset representatives to compute traces without enumerating all group elements. For example, inducing the trivial representation of H yields the permutation character on G/H, with \chi^G(g) counting fixed cosets under conjugation by g.^[34] In the context of reductive groups like \mathrm{GL}(n), Hecke algebras are algebras generated by double cosets that act on representations. For G = \mathrm{GL}(n, \mathbb{Q}_p) and a hyperspecial maximal compact subgroup K = \mathrm{GL}(n, \mathbb{Z}_p), the spherical Hecke algebra \mathcal{H}(G, K) consists of compactly supported bi-K-invariant functions under convolution, with a \mathbb{Z}-basis given by the characteristic functions of double cosets K \gamma K for \gamma \in G.^[37] These double cosets are parametrized by dominant cocharacters (e.g., diagonal matrices with non-increasing p-adic valuations), and the algebra is commutative, isomorphic to a polynomial ring in n variables over \mathbb{C}, facilitating the study of unramified principal series representations of \mathrm{GL}(n).^[37] The Iwahori-Hecke algebra, arising from double cosets modulo the Iwahori subgroup (parahoric stabilizer of a Borel), deforms the Weyl group algebra and governs the Bernstein components in the representation theory of \mathrm{GL}(n).^[37]

History

Origins in Number Theory

The ideas underlying cosets originated in the study of congruences and residue classes within number theory during the 18th and 19th centuries. Mathematicians such as Leonhard Euler and Joseph-Louis Lagrange employed modular arithmetic in their work on Diophantine equations and Fermat's Little Theorem during the 1760s and 1770s. Lagrange, in his 1771 memoir "Réflexions sur la résolution algébrique des équations," used remainders from division (modulo divisors) to reduce coefficients in polynomial equations, providing early insights into partitioning integers based on congruence.^[38] These approaches laid groundwork for understanding the additive group of integers divided into disjoint subsets of equal size.^[39] Carl Friedrich Gauss formalized these notions in his seminal 1801 work Disquisitiones Arithmeticae, where he defined congruences a \equiv b \pmod{m} as the condition that m divides a - b, explicitly describing how such congruences partition the integers into m distinct residue classes, each corresponding to a coset in the additive group \mathbb{Z}.^[40] Gauss's framework emphasized the uniformity of these classes, noting their role in solving Diophantine equations and proving unique factorization, which highlighted the partitioning property central to coset theory.^[41] In the 1810s, Augustin-Louis Cauchy advanced early ideas of group structure through his investigations of permutations, where concepts akin to cosets emerged implicitly in the analysis of symmetries and substitutions acting on sets.^[42] Cauchy's work on the symmetric group of permutations laid the foundation for viewing permutations as generating equivalence relations that divide the set into orbits, foreshadowing coset decompositions in more abstract settings.^[43] Leonhard Euler's earlier contributions, particularly his introduction of the totient function \phi(n) in the 1760s, connected to the count of integers up to n that are coprime to n, which represents the number of units in the multiplicative group modulo n and prefigures the notion of index as the size ratio between a group and its subgroup.^[44] This function, derived from Euler's studies of cyclotomic polynomials and Fermat's Little Theorem, underscored the finite structure of residue systems, influencing later developments in coset-based partitioning.^[44]

Development in Abstract Algebra

Évariste Galois played a pivotal role in the early development of abstract group theory in the 1830s, introducing the concept of cosets in his manuscripts on the solvability of polynomial equations by radicals. Galois decomposed permutation groups into cosets relative to subgroups to study the structure of equation roots under substitutions, establishing that the number of distinct cosets equals the index of the subgroup—a key idea for understanding group actions and normal subgroups. His work, published posthumously in 1846, bridged concrete permutation groups and abstract structures, directly influencing later formalizations of cosets.^[45] The development of cosets within abstract group theory continued in the mid-19th century, as mathematicians shifted from concrete permutation representations to more general structures. Arthur Cayley laid foundational work in his 1854 paper "On the theory of groups, as depending on the symbolic equation θ^n = 1," where the concept of cosets appeared implicitly in the definition of groups through relations and symmetries. By the 1870s, Cayley made explicit use of cosets in enumerating finite groups, employing them to partition group elements and analyze subgroups in works such as his 1869 memoir on cubic surfaces and his 1878 papers on group theory.^[43] Camille Jordan advanced this framework significantly in his 1870 treatise Traité des substitutions et des équations algébriques, where he introduced cosets explicitly within the context of symmetric groups. Jordan used cosets to decompose permutation groups into disjoint classes relative to subgroups, facilitating the study of transitive actions and subgroup indices in finite symmetric groups.^[46] This formalization marked a key step toward abstracting cosets beyond specific numerical examples, emphasizing their role in group decompositions. William Burnside further centralized cosets in his 1897 book Theory of Groups of Finite Order, treating them as essential tools for proving the Sylow theorems. In these theorems, cosets partition the group to count Sylow subgroups and establish their congruence properties modulo the subgroup order, providing a cornerstone for the classification of finite groups.^[47] In the 20th century, Emil Artin integrated cosets into the broader structure of quotient groups and group actions during the 1940s, notably in his lectures on Galois Theory (1944). Artin showed how cosets of normal subgroups form the elements of quotient groups, enabling homomorphic images and extensions central to solvability criteria. Cosets are particularly significant when the subgroup is normal, as this condition ensures the cosets multiply associatively to yield a group structure. Post-1950 developments extended cosets to infinite and topological settings, such as profinite groups, where open subgroups have finitely many cosets that are also open sets in the profinite topology. Jean-Pierre Serre formalized this in Cohomologie galoisienne (1964), using cosets to describe continuous actions and Galois correspondences in infinite extensions. More recently, from the 1970s onward, cosets have found applications in algebraic geometry through étale groups, particularly the étale fundamental group. Alexander Grothendieck's Revêtements étales et groupe fondamental (SGA 1, 1971) employs cosets to classify finite étale covers via orbits under group actions, linking them to Galois representations and anabelian geometry.

References

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Every element of G belongs to exactly one right coset of H in G. Thus, G is the disjoint union of the distinct left cosets of H in G. Also, G is the disjoint ...
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[PDF] Lecture 13 - Math 5111 (Algebra 1)
Oct 19, 2020 · Definition. If H is a subgroup of G and a ∈ G, the set aH = {ah : h ∈ H} is called a left coset of H. We also define the index of H in G, ...
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[PDF] Cosets and Lagrange's theorem - Keith Conrad
This is similar to Example 2.1, where the cosets are a family of parallel lines: different parallel lines are disjoint.
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[PDF] Cyclic groups. Cosets. Lagrange's theorem.
There are only 2 cosets, the set of even permutations A(n) and the set of odd permutations S(n) \ A(n). • G is any group, H = G. There is only one coset, G. •
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[PDF] 3 Basic concepts in group theory - 3.2 Subgroup - Xie Chen
Coset, definition: given a subgroup H = {h1,h2, ..., hr} of a group G, the left coset given by an. element g ∈ G, written as gH, is defined as the set of ...
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Coset -- from Wolfram MathWorld
A subset of G of the form xH for some x in G is said to be a left coset of H and a subset of the form Hx is said to be a right coset of H.
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https://mathworld.wolfram.com/Coset.html
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https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_(Ernst)/05%3A_Cosets_Lagranges_Theorem_and_Normal_Subgroups/5.01%3A_Cosets
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[PDF] 10 Cosets and Lagrange's Theorem - UC Berkeley math
Recall that every equivalence relation on a set S partitions S into equivalence classes. For the equivalence relation ∼L, what is the equivalence class for an ...
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[PDF] groups.pdf - Dexter Chua
Group theory is an example of algebra. ... Before we prove these, we first study these subgroups a bit. 4.1 Normal subgroups. Definition (Normal subgroup).<|control11|><|separator|>
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normal subgroup - PlanetMath.org
Mar 22, 2013 · A subgroup H H of a group G G is normal if aH=Ha a ⁢ H = H ⁢ a for all a∈G a ∈ G . Equivalently, H⊂G H ⊂ G is normal if and only if aHa−1=H ...
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https://planetmath.org/normalsubgroup
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[PDF] Math 4310 Handout - Quotient Vector Spaces - Cornell Mathematics
Then the equivalence class of a vector v under the relation ∼ above is the set v + W = {v + w : w ∈ W}; this is called an affine subspace a coset for v and W.
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[PDF] 1. Cosets and the Quotient Space - Christopher Heil
Since two cosets of M are either identical or disjoint, the quotient space X/M is the set of all the distinct cosets of M. 2/M = {(0,y2) + M : y2 ∈ R}.
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[PDF] NOTES ON QUOTIENT SPACES Let V be a vector ... - Academic Web
Let V be a vector space over a field F, and let W be a subspace of V . There is a sense in which we can “divide” V by W to get a new vector space.
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Linear Algebra, Part 3: Quotient Spaces (Mathematica)
The associated equivalence classes are called cosets or affine k-planes. The quotient V/W forms a vector space with dimension dim(V) − dim(W).<|control11|><|separator|>
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[PDF] Advanced constructions of vector spaces.
A coset of V0 in. V is any set of the form {x} + V0 (also denoted x + V0). The set of all cosets of V0 is denoted V/V0 and called the quotient of V by V0.
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[PDF] Algebraic Groups - James Milne
Dec 20, 2015 · An algebraic group is a matrix group defined by polynomial conditions. More abstractly, it is a group scheme of finite type over a field.
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[PDF] LECTURE II 1. General Linear Group Let Fq be a finite field of order ...
Example: Let p be a prime and G = GL3(p). Let U be the group of upper triangular matrices with 1 on the diagonal (so called unipotent matrices), and let T be ...
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[PDF] group actions - keith conrad
When a group G acts on G/H (for a subgroup H) by left multiplication,. • there is one orbit (gH = g · H ∈ OrbH),. • StabaH = {g : gaH = aH} = {g : a−1ga ∈ H} = ...<|control11|><|separator|>
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https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Introduction_to_Algebraic_Structures_(Denton)/06%3A_Group_Actions/6.02%3A_Orbits_and_Stabilizers
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[PDF] Abstract Algebra
Since left cosets are orbits under the action of H on G by right multiplication, the lemma follows by Exercise 15.11. For a change, here is an explicit ...
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[PDF] Abstract Algebra
... Page 5. ABSTRACT ALGEBRA. Third Edition. David S. Dummit. University of Vermont. Richard M. Foote. University of Vermont john Wiley & Sons, Inc. Page 6 ...
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[PDF] Group Theory - James Milne
These notes give a concise exposition of the theory of groups, including free groups and Coxeter groups, the Sylow theorems, and the representation theory ...
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[PDF] More on induced representations - Columbia Math Department
Finally, note that every double coset KxH is a union of left cosets of H ... where we have used Frobenius reciprocity twice and Mackey's theorem to write ...
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[PDF] Cosets, Syndromes, and Standard Arrays - UCSD CSE
Error detection: Compute the syndrome s, check if s 6= 0. Error correction ... Decoding procedure: Keep a table of (syndrome, coset leader). The.
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[PDF] MATH 433 Applied Algebra Lecture 32: Linear codes. Coset leaders ...
Using coset leaders, the error correction can be done via the coset decoding table, which is a 2n-m×2m table containing all words of length m. The table has the ...
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[PDF] Linear Codes and Syndrome Decoding
Since the cosets are in a one-to-one correspondence with the syndromes, we can also do this using a smaller table with two columns in which the first column ...
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[PDF] The Bell System Technical Journal - Zoo | Yale University
April, 1950. The Bell System Technical Journal. Vol. xx~. Copyright, 1950,American Telephone and Telegraph Company. Error Detecting and Error Correcting Codes.
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[PDF] Hamming codes
We give a construction of a q-ary Hamming code and prove that it is perfect with minimum distance 3. We show that syndrome decoding works for Hamming codes in ...
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[PDF] Lecture 12. Induced Characters
May 19, 2020 · To prove this, we note that the degree d of the representation with character χ is d = χ(1). So the degree χG(1) of the induced representation ...
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[PDF] Induced representations - Columbia Math Department
{1} C = L 2(G) = C[G], the regular representation. functions F : G → C which are constant on the cosets gH, for every g ∈ G. xH).
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[PDF] Math 210B. Mackey theory and applications
where s is a representative of the double coset s ... Serre's book Linear representations of finite groups via Mackey's criterion and Frobenius reciprocity.
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[PDF] Hecke Algebras
May 11, 2010 · The Iwahori Hecke algebra is closely related to the Temperly-Lieb algebras which arise in both statistical physics and quantum physics. The ...
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[PDF] A History of Lagrange's Theorem on Groups
Section 9 of chapter VI is entitled. "Lagrange's Theorem." It started with two lemmas showing that each coset has the same number of elements as the subgroups ...
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[PDF] Algebra and Number Theory1
Oct 22, 2000 · 2.1 Residues. Euler also considered residue classes: a = r(mod d) ... Proved in 1771 by Lagrange in 1771 In his proof, he noted that the.
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Disquisitiones Arithmeticae | Carl Friedrich GAUSS | First edition
He introduced congruence of integers with respect to a modulus (a ≡ b (mod c) if c divides a - b), the first significant algebraic example of the now ...Missing: partitioning | Show results with:partitioning
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Gauss: "Disquisitiones Arithmeticae" - MacTutor
In the year 1801 seven sections of the Disquisitiones Arithmeticae were published in octavo. The first reprint was published under my direction in 1863 as the ...Missing: partitioning | Show results with:partitioning
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[PDF] Early Group Theory in the Work of A.-L. Cauchy
Jul 1, 2025 · • The symmetric group Sn is the set of all permutations on n objects.22. – This is what Cauchy called (somewhat long-windedly):. 'the system ...
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[PDF] Early group theory in the works of Lagrange, Cauchy, and Cayley
Aug 19, 2010 · • The symmetric group Sn is the set of all permutations on n objects.46 ... cosets of H that corresponded to permutations selected from outside of ...
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Math Origins: The Totient Function
Leonhard Euler's totient function, φ(n), is an important object in number theory, counting the number of positive integers less than or equal to n which are ...Missing: modulo | Show results with:modulo
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Traité des substitutions et des équations algébriques - Internet Archive
Dec 3, 2008 · Traité des substitutions et des équations algébriques. by: Jordan, Camille, 1838-1922. Publication date: 1870. Topics: Galois theory, Groups ...Missing: cosets introduction
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Theory of groups of finite order : Burnside, William, 1852-1927
Nov 8, 2009 · Theory of groups of finite order : Burnside, William, 1852-1927 : Free Download, Borrow, and Streaming : Internet Archive.