# Coset

A **coset** is a translate of a subgroup inside a larger group: given a group \( G \), a subgroup \( H \leq G \), and an element \( g \in G \), the left coset \( gH \) is the set \( \{ gh \mid h \in H \} \) and the right coset \( Hg \) is the set \( \{ hg \mid h \in H \} \)<sup>[1](https://math.umd.edu/~pbrosnan/notes/ugalg/sect0019.html)</sup>. Cosets are the equivalence classes of a natural relation on \( G \), they partition \( G \) into equal-sized blocks, and their counting properties yield Lagrange's theorem. When the subgroup is normal, meaning its left and right cosets coincide, the set of cosets itself forms a group called the [quotient group](https://www.edgepedia.org/quotient-group)<sup>[2](https://sites.ualberta.ca/~vbouchar/MAPH464/section-quotient.html)</sup>.

| Key fact | Detail |
|---|---|
| Definition | Left coset \( gH = \{ gh : h \in H \} \); right coset \( Hg = \{ hg : h \in H \} \); \( G/H \) denotes the set of all left cosets<sup>[1](https://math.umd.edu/~pbrosnan/notes/ugalg/sect0019.html)</sup> |
| Partition | The left cosets of \( H \) form a partition of \( G \), as the equivalence classes of the relation \( g_1 \sim g_2 \iff g_1^{-1}g_2 \in H \)<sup>[1](https://math.umd.edu/~pbrosnan/notes/ugalg/sect0019.html)</sup> |
| Equal size | The map \( \phi(h) = gh \) is a bijection from \( H \) to \( gH \), so every coset has the same size as \( H \)<sup>[3](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_%28Ernst%29/05%3A_Cosets_Lagranges_Theorem_and_Normal_Subgroups/5.01%3A_Cosets)</sup> |
| Index | The index \( [G:H] \) is the number of left cosets of \( H \) in \( G \); for finite groups \( [G:H] = |G|/|H| \)<sup>[1](https://math.umd.edu/~pbrosnan/notes/ugalg/sect0019.html)</sup><sup> • </sup><sup>[2](https://sites.ualberta.ca/~vbouchar/MAPH464/section-quotient.html)</sup> |
| Normality | A subgroup \( N \) is normal if and only if \( aN = Na \) for all \( a \in G \)<sup>[4](https://mathresearch.utsa.edu/wiki/index.php?title=Normal_Subgroups_and_Factor_Groups)</sup> |
| Quotient group | If \( N \) is normal, the cosets form a group \( G/N \) under \( (aN)(bN) = abN \)<sup>[5](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%253A_Theory_and_Applications_%28Judson%29/10%253A_Normal_Subgroups_and_Factor_Groups/10.01%253A_Factor_Groups_and_Normal_Subgroups)</sup> |
| Isomorphism theorem | For a homomorphism \( \varphi \) with kernel \( K \), the map \( \Phi(aK) = \varphi(a) \) is an isomorphism \( G/K \cong \varphi(G) \)<sup>[6](https://www.math.uwaterloo.ca/~snew/PMATH336/Chap5QuotientGroups.pdf)</sup> |

## Definition and basic properties

Fix a group \( G \) with identity element \( e \) and a subgroup \( H \). The coset \( H = eH = He \) is always both a left and a right coset, called the trivial coset<sup>[1](https://math.umd.edu/~pbrosnan/notes/ugalg/sect0019.html)</sup>. Any element of a coset may serve as its **coset representative**: \( b \in aH \) if and only if \( aH = bH \)<sup>[3](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_%28Ernst%29/05%3A_Cosets_Lagranges_Theorem_and_Normal_Subgroups/5.01%3A_Cosets)</sup>. Equivalently, two elements \( g_1, g_2 \) lie in the same left coset exactly when \( g_1^{-1}g_2 \in H \)<sup>[1](https://math.umd.edu/~pbrosnan/notes/ugalg/sect0019.html)</sup>.

Three structural facts follow from the subgroup property. First, the relation \( g_1 \sim_L g_2 \iff g_1^{-1}g_2 \in H \) is reflexive, symmetric, and transitive, so its equivalence classes partition \( G \), and those classes are precisely the left cosets \( gH \)<sup>[1](https://math.umd.edu/~pbrosnan/notes/ugalg/sect0019.html)</sup>. Second, left multiplication by a fixed \( g \) is a bijection from \( H \) onto \( gH \), so \( \#(aH) = |H| \) for every \( a \), and the same holds for right cosets; the argument works even when \( H \) has infinite order<sup>[3](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_%28Ernst%29/05%3A_Cosets_Lagranges_Theorem_and_Normal_Subgroups/5.01%3A_Cosets)</sup>. Third, distinct cosets are disjoint, since two cosets that share one element are identical<sup>[3](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_%28Ernst%29/05%3A_Cosets_Lagranges_Theorem_and_Normal_Subgroups/5.01%3A_Cosets)</sup>.

Among all cosets, only the subgroup itself is a subgroup: \( H \) is the only left or right coset of \( H \) that is a subgroup of \( G \)<sup>[7](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_%28Sklar%29/07%3A_The_Wonderful_World_of_Cosets/7.02%3A_Introduction_to_Cosets_and_Normal_Subgroups)</sup>.

In additive notation, common for abelian groups, the coset generated by \( a \) is written \( a + H = \{ a + h \mid h \in H \} \)<sup>[8](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Elementary_Abstract_Algebra_%28Clark%29/01%3A_Chapters/1.08%3A_Cosets_and_Lagrange%27s_Theorem)</sup>.

## Left versus right cosets

Left and right cosets of the same subgroup need not coincide. If \( G \) is abelian, then \( aH = Ha \) for all \( a \in G \), so the distinction disappears<sup>[3](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_%28Ernst%29/05%3A_Cosets_Lagranges_Theorem_and_Normal_Subgroups/5.01%3A_Cosets)</sup>. In non-abelian groups the cosets of a subgroup may or may not coincide; knowing only that the group is non-abelian does not determine whether the left and right cosets differ<sup>[3](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_%28Ernst%29/05%3A_Cosets_Lagranges_Theorem_and_Normal_Subgroups/5.01%3A_Cosets)</sup>.

A subgroup \( N \) for which \( aN = Na \) for every \( a \in G \) is called a **normal subgroup**<sup>[4](https://mathresearch.utsa.edu/wiki/index.php?title=Normal_Subgroups_and_Factor_Groups)</sup>. For a normal subgroup the left and right cosets agree, so one can speak of cosets without specifying a side<sup>[2](https://sites.ualberta.ca/~vbouchar/MAPH464/section-quotient.html)</sup>.

## Lagrange's theorem and the index

Because the cosets partition \( G \) and all have the same size as \( H \), a finite group of order \( n \) splits into \( s \) disjoint cosets \( a_1H, \dots, a_sH \), each of size \( |H| \), giving \( |G| = s \cdot |H| \)<sup>[8](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Elementary_Abstract_Algebra_%28Clark%29/01%3A_Chapters/1.08%3A_Cosets_and_Lagrange%27s_Theorem)</sup>. This is Lagrange's theorem: the order of a subgroup divides the order of the group<sup>[9](https://fmwww.bc.edu/gross/MT216/aata.pdf)</sup>.

The number of distinct cosets is the **index** \( [G:H] \), written as \( \infty \) when that number is infinite<sup>[1](https://math.umd.edu/~pbrosnan/notes/ugalg/sect0019.html)</sup>. For a finite group, \( [G:H] = |G|/|H| \)<sup>[2](https://sites.ualberta.ca/~vbouchar/MAPH464/section-quotient.html)</sup>.

A worked example shows how the count works. In \( U_{13} \), the group of units modulo 13, the element \( 2 \) has order 12, so \( 2^4 \) has order 3 and \( H = \langle 2^4 \rangle = \{ 1, 2^4, 2^8 \} \) is a subgroup. Listing \( 2H, 2^2H, \dots, 2^{12}H \) appears to give twelve cosets, but repetitions collapse them to exactly four distinct cosets, matching \( [U_{13}:H] = 12/3 = 4 \)<sup>[8](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Elementary_Abstract_Algebra_%28Clark%29/01%3A_Chapters/1.08%3A_Cosets_and_Lagrange%27s_Theorem)</sup>.

## Quotient groups

Normality is exactly the condition that lets the set of cosets inherit a group structure. If \( N \) is normal in \( G \), the collection of distinct cosets \( G/N \) forms a group under the product of subsets, with \( (aN)(bN) = abN \)<sup>[2](https://sites.ualberta.ca/~vbouchar/MAPH464/section-quotient.html)</sup><sup> • </sup><sup>[5](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%253A_Theory_and_Applications_%28Judson%29/10%253A_Normal_Subgroups_and_Factor_Groups/10.01%253A_Factor_Groups_and_Normal_Subgroups)</sup>. Normality is what makes the operation well-defined, that is, independent of the chosen coset representatives: an element of \( (aH)(bH) \) has the form \( a h_1 b h_2 \), and since \( H \) is normal one can write \( h_1 b = b h_3 \) for some \( h_3 \in H \), giving \( a h_1 b h_2 = (ab) h_3 h_2 \in (ab)H \)<sup>[2](https://sites.ualberta.ca/~vbouchar/MAPH464/section-quotient.html)</sup>. The identity is the coset \( eH = H \), and the inverse of \( gH \) is \( g^{-1}H \)<sup>[2](https://sites.ualberta.ca/~vbouchar/MAPH464/section-quotient.html)</sup>. The resulting group \( G/N \) is called the quotient group or factor group of \( G \) by \( N \)<sup>[5](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%253A_Theory_and_Applications_%28Judson%29/10%253A_Normal_Subgroups_and_Factor_Groups/10.01%253A_Factor_Groups_and_Normal_Subgroups)</sup>.

Cosets also connect quotient groups to homomorphisms. The **First Isomorphism Theorem** states that if \( \varphi : G \to H \) is a group homomorphism with kernel \( K \), then \( K \) is normal and \( G/K \cong \varphi(G) \) via \( \Phi(aK) = \varphi(a) \); conversely, for any normal \( K \), the map \( \varphi(a) = aK \) is a homomorphism \( G \to G/K \) with kernel \( K \)<sup>[6](https://www.math.uwaterloo.ca/~snew/PMATH336/Chap5QuotientGroups.pdf)</sup>. In a quotient, the equivalence class of the identity is a normal subgroup of the original group, and the other classes are precisely its cosets<sup>[4](https://mathresearch.utsa.edu/wiki/index.php?title=Normal_Subgroups_and_Factor_Groups)</sup>.

## Cosets in group actions and applications

Cosets appear wherever a group acts on a set. The **Orbit-Stabilizer theorem** gives a bijection between the elements of the orbit \( \omega^G \) and the cosets of the stabilizer \( \mathrm{Stab}_G(\omega) \), so that \( |G| = |\omega^G| \cdot |\mathrm{Stab}_G(\omega)| \); in particular, the length of an orbit divides the order of the acting group<sup>[10](https://www.math.colostate.edu/~hulpke/lectures/m502/fa20.pdf)</sup>.

Specialized coset constructions carry the name in other areas. The **cyclotomic cosets** modulo \( n \) are the equivalence classes of the relation \( b = aq^i \) on \( \{0, 1, \dots, n-1\} \) for relatively prime \( q \) and \( n \); a cyclotomic coset has the form \( \{ s, sq, sq^2, \dots \} \), and these classes are used to construct \( q \)-ary cyclic codes of length \( n \)<sup>[11](https://mypage.cuhk.edu.cn/academics/wkshum/sage/cyclotomic_coset.html)</sup>. **Double cosets** of the form \( \Gamma \alpha \Gamma \), unions of left and right cosets of a subgroup \( \Gamma \), form the basis of the [Hecke algebra](https://www.edgepedia.org/hecke-algebra) \( R[\Gamma \backslash \Delta / \Gamma] \), an \( R \)-algebra built from double cosets of a subgroup inside a semigroup \( \Delta \)<sup>[12](https://math.mit.edu/classes/18.786/2024/LectureNotes10.pdf)</sup>.

Cosets also have a visual reading: in a Cayley diagram of a finite group, the clones of a subgroup, the copies of the subgroup's structure reached by the same generators, coincide with the right cosets of that subgroup<sup>[3](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_%28Ernst%29/05%3A_Cosets_Lagranges_Theorem_and_Normal_Subgroups/5.01%3A_Cosets)</sup>.

## History

The coset idea predates its modern formulation. In 1761, Euler's work on modular arithmetic, though not stated in group-theoretic terms, provides an example of the decomposition of an abelian group into cosets of a subgroup, and he proved a special case of the order of a subgroup dividing the order of the group<sup>[13](https://mathshistory.st-andrews.ac.uk/HistTopics/Development_group_theory/)</sup>. [Évariste Galois](https://www.edgepedia.org/evariste-galois), by 1832, had recognized that special subgroups, now called normal subgroups, are fundamental, and he called the decomposition of a group into cosets of a subgroup a *proper decomposition* when the right and left coset decompositions coincide<sup>[13](https://mathshistory.st-andrews.ac.uk/HistTopics/Development_group_theory/)</sup>.

The attribution of the decomposition itself shifted over time. The rectangular array often used to display a coset decomposition is usually credited to [Augustin-Louis Cauchy](https://www.edgepedia.org/augustin-louis-cauchy), while the coset decomposition equation frequently found in the literature is due to Galois; the two threads were brought together in Heinrich Weber's book. Bartel van der Waerden's *Moderne Algebra* (1930), one of the most influential algebra texts, proved Lagrange's theorem by the coset approach without mentioning Lagrange's name, though a footnote credited the coset decomposition to Galois; the 1937 revised edition explicitly stated and proved that any two cosets have the same number of elements<sup>[14](https://www.math.univ-toulouse.fr/~bauval/Roth.pdf)</sup>.

## References

1. [UMD MATH403: Cosets and Lagrange's Theorem (Definitions 2.84–2.87)](https://math.umd.edu/~pbrosnan/notes/ugalg/sect0019.html)
2. [Quotient groups, MAPH 464, University of Alberta](https://sites.ualberta.ca/~vbouchar/MAPH464/section-quotient.html)
3. [5.1: Cosets, An Inquiry-Based Approach to Abstract Algebra (Ernst), LibreTexts](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/An_Inquiry-Based_Approach_to_Abstract_Algebra_%28Ernst%29/05%3A_Cosets_Lagranges_Theorem_and_Normal_Subgroups/5.01%3A_Cosets)
4. [Normal Subgroups and Factor Groups, UTSA Mathematics Wiki](https://mathresearch.utsa.edu/wiki/index.php?title=Normal_Subgroups_and_Factor_Groups)
5. [10.1: Factor Groups and Normal Subgroups, Abstract Algebra: Theory and Applications (Judson), LibreTexts](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Abstract_Algebra%253A_Theory_and_Applications_%28Judson%29/10%253A_Normal_Subgroups_and_Factor_Groups/10.01%253A_Factor_Groups_and_Normal_Subgroups)
6. [Chapter 5: Cosets, Normal Subgroups, and Quotient Groups, PMATH 336, University of Waterloo](https://www.math.uwaterloo.ca/~snew/PMATH336/Chap5QuotientGroups.pdf)
7. [7.2: Introduction to Cosets and Normal Subgroups (Sklar), LibreTexts](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/First-Semester_Abstract_Algebra%3A_A_Structural_Approach_%28Sklar%29/07%3A_The_Wonderful_World_of_Cosets/7.02%3A_Introduction_to_Cosets_and_Normal_Subgroups)
8. [1.08: Cosets and Lagrange's Theorem (Clark), LibreTexts](https://math.libretexts.org/Bookshelves/Abstract_and_Geometric_Algebra/Elementary_Abstract_Algebra_%28Clark%29/01%3A_Chapters/1.08%3A_Cosets_and_Lagrange%27s_Theorem)
9. [Abstract Algebra (MT 216 course text, Boston College)](https://fmwww.bc.edu/gross/MT216/aata.pdf)
10. [Orbits and Stabilizers, M502 lecture notes, A. Hulpke, Colorado State University](https://www.math.colostate.edu/~hulpke/lectures/m502/fa20.pdf)
11. [Cyclotomic cosets, W.-K. Shum, The Chinese University of Hong Kong, Shenzhen](https://mypage.cuhk.edu.cn/academics/wkshum/sage/cyclotomic_coset.html)
12. [18.786 Number Theory II, Lecture 10, MIT (Hecke algebras and double cosets)](https://math.mit.edu/classes/18.786/2024/LectureNotes10.pdf)
13. [Development of Group Theory, MacTutor History of Mathematics](https://mathshistory.st-andrews.ac.uk/HistTopics/Development_group_theory/)
14. [A History of Lagrange's Theorem on Groups (Roth), Université de Toulouse](https://www.math.univ-toulouse.fr/~bauval/Roth.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Group theory › Group structures and subgroups*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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