# Group ring

A **group ring** is a ring formed from a ring R and a group G by taking formal finite sums of group elements with coefficients in R, so that an element of R[G] is a sum \( \sum_{g} r_g g \) with only finitely many nonzero coefficients, and multiplication extends the group operation by the rule \( [g][h] = [gh] \). When R is a field, R[G] is called the **group algebra** of G over R.<sup>[1](https://alphabetamath.cc/item/def-group-ring)</sup><sup> • </sup><sup>[2](https://math.uni.lu/kiefer/Publications/thesis.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | R[G] is the free R-module on the set G; every element is a unique finite sum \( \sum r_g [g] \), with multiplication defined by \( [g][h] = [gh] \)<sup>[1](https://alphabetamath.cc/item/def-group-ring)</sup> |
| Generality | The same construction works when G is only a semigroup, giving the semigroup ring R[G]<sup>[3](https://users.math.msu.edu/users/meierfra/Classnotes/M818F12/M818F12notes.pdf)</sup> |
| Size | If G is finite, the group algebra k[G] has dimension |G| over the field k<sup>[1](https://alphabetamath.cc/item/def-group-ring)</sup> |
| Commutativity | k[G] is commutative if and only if G is abelian<sup>[4](https://www.math.nagoya-u.ac.jp/~richard/teaching/s2015/Khalkhali_13.pdf)</sup> |
| Hopf structure | The group algebra is a Hopf algebra with \( \Delta(g) = g \otimes g \), \( \varepsilon(g) = 1 \), and antipode \( S(g) = g^{-1} \)<sup>[5](https://utensil.github.io/forest/fgap-001V/)</sup> |
| Representation theory | Representations of a finite group G over a field k are the same objects as left kG-modules<sup>[6](https://kevos.com/engineering/group-representations-and-modules)</sup> |
| Kaplansky unit conjecture | Disproved in 2021 by Giles Gardam with a virtually abelian group over the field of order two<sup>[7](https://annals.math.princeton.edu/2021/194-3/p09)</sup> |

## Definition and first properties

Let R be a commutative ring and G a group. The group ring R[G] is the free left R-module on the set G: each group element g gives a basis vector [g], and every element has a unique expression \( \sum_{g \in F} r_g [g] \) with F a finite subset of G and coefficients in R. The support is finite by construction because R[G] is the direct sum, not the full product, of copies of R indexed by G.<sup>[1](https://alphabetamath.cc/item/def-group-ring)</sup>

Multiplication is determined on basis vectors by \( [g][h] = [gh] \) and extended R-linearly. With this product R[G] is a unital R-algebra whose basis is G itself, and each basis vector [g] is a unit of the ring, with inverse \( [g^{-1}] \).<sup>[1](https://alphabetamath.cc/item/def-group-ring)</sup> In full generality the construction only needs G to be a semigroup: the semigroup ring R[G] is defined as the direct sum of copies of R indexed by G, with multiplication given by the convolution formula \( (rs)_g = \sum_{kl=g} r_k s_l \), which is well defined because supports are finite. If R and G have identities, then \( 1_R 1_G \) is the identity of R[G], and if R and G are commutative, so is R[G].<sup>[3](https://users.math.msu.edu/users/meierfra/Classnotes/M818F12/M818F12notes.pdf)</sup>

The notation deliberately parallels that of polynomial rings: a group ring in a cyclic group behaves like a ring of Laurent-type expressions in one generator, and the formal-sum viewpoint makes the algebra structure transparent.<sup>[1](https://alphabetamath.cc/item/def-group-ring)</sup>

## The Hopf algebra structure

A group algebra carries more structure than an ordinary algebra. With comultiplication \( \Delta([g]) = [g] \otimes [g] \), counit \( \varepsilon([g]) = 1 \), and antipode \( S([g]) = [g^{-1}] \), the group algebra k[G] becomes a [Hopf algebra](https://www.edgepedia.org/hopf-algebra), an algebra equipped with algebra homomorphisms Δ and ε and a linear map S satisfying axioms that dualize the associativity, unit, and inverse axioms of a group.<sup>[5](https://utensil.github.io/forest/fgap-001V/)</sup><sup> • </sup><sup>[4](https://www.math.nagoya-u.ac.jp/~richard/teaching/s2015/Khalkhali_13.pdf)</sup> The antipode extends inversion on G and satisfies the two convolution identities.<sup>[5](https://utensil.github.io/forest/fgap-001V/)</sup>

This structure is not decorative. The original group G can be recovered from the Hopf algebra kG as its set of grouplike elements, those nonzero h satisfying \( \Delta(h) = h \otimes h \); in any Hopf algebra a grouplike element is invertible with inverse S(g), and the grouplike elements form a group.<sup>[4](https://www.math.nagoya-u.ac.jp/~richard/teaching/s2015/Khalkhali_13.pdf)</sup><sup> • </sup><sup>[8](https://notes.edubirdie.com/docs/massachusetts-institute-of-technology/18-769-topics-in-lie-theory-tensor-ca/86435-18-769-lecture-5-notes)</sup> The group algebra is cocommutative, and it is commutative if and only if G is commutative.<sup>[4](https://www.math.nagoya-u.ac.jp/~richard/teaching/s2015/Khalkhali_13.pdf)</sup> When G is finite there is a second Hopf algebra attached to G, the dual algebra (kG)* of functions on G, and the two are dual to each other; the group algebra kG is a pointed cosemisimple cocommutative Hopf algebra, and (kG)* is commutative and semisimple.<sup>[4](https://www.math.nagoya-u.ac.jp/~richard/teaching/s2015/Khalkhali_13.pdf)</sup><sup> • </sup><sup>[9](https://www.sciencedirect.com/topics/mathematics/hopf-algebras)</sup>

## Group rings and representation theory

The principal reason group rings matter is that they convert group representation problems into module problems. For a finite group G and a field k, a k-representation of G and a left kG-module are the same object described twice: the assignment sending a representation \( \rho \) to the induced module structure on V is a bijection that preserves every structural notion on both sides.<sup>[6](https://kevos.com/engineering/group-representations-and-modules)</sup> Concretely, if \( \rho : G \to \mathrm{GL}(V) \) is a linear representation, setting \( sx = \rho_s x \) for \( s \in G \) and extending linearly makes V a left K[G]-module, and conversely any such module structure defines a representation.<sup>[10](https://www.math.tau.ac.il/~borovoi/courses/ReprFG/Hatzagot.pdf)</sup>

**Semisimplicity.** Maschke's theorem states that if G is finite and F is a field whose characteristic does not divide |G|, then every subrepresentation of an F[G]-module is a direct summand; equivalently, the group algebra F[G] is a semisimple [F-algebra](https://www.edgepedia.org/f-algebra), isomorphic to a finite direct product of matrix rings over finite-dimensional division algebras over F.<sup>[11](https://math.ucla.edu/~sharifi/notes/algebra-ch13.html)</sup> When F is algebraically closed, F[G] is a direct product of matrix algebras over F, and the number of factors equals the number of conjugacy classes of G.<sup>[11](https://math.ucla.edu/~sharifi/notes/algebra-ch13.html)</sup> Over a field of characteristic zero, two finite-dimensional representations are isomorphic exactly when their characters agree.<sup>[11](https://math.ucla.edu/~sharifi/notes/algebra-ch13.html)</sup>

When the characteristic of the field divides the group order, the situation changes: modules need not be semisimple, and for a cyclic group of order \( p^n \) over a field of characteristic p there is an isomorphism \( kG \cong k[X]/(X^{p^n}) \), a non-semisimple ring.<sup>[12](https://www-users.cse.umn.edu/~webb/RepBook/RepBookLatex.pdf)</sup>

## Units and Kaplansky's conjectures

The unit group of a group ring is a central object of study. For the integral group ring \( \mathbb{Z}G \) of a finite group G, three decades of work have produced constructions of large torsion-free subgroups, meaning subgroups of finite index, inside the unit group \( \mathcal{U}(\mathbb{Z}G) \); these constructions rely on explicit units in \( \mathbb{Z}G \) and on the description of the Wedderburn components of the rational group algebra \( \mathbb{Q}G \).<sup>[13](https://arxiv.org/abs/2008.11569)</sup> The study of finite subgroups of the unit group of an integral group ring was initiated by [Graham Higman](https://www.edgepedia.org/graham-higman) in the 1940s, and major questions in this area, such as the Isomorphism Problem and the Zassenhaus Conjectures, have been settled.<sup>[14](https://cris.technion.ac.il/en/publications/finite-subgroups-of-group-rings-a-survey/)</sup>

**The unit conjecture.** Kaplansky's unit conjecture predicted that if K is a field and G is a torsion-free group, then the only units of K[G] are the trivial units, the nonzero scalar multiples of group elements. In 2021 Giles Gardam, a mathematician then at the Mathematisches Institut of the Universität Münster, published a concrete counterexample in the [Annals of Mathematics](https://www.edgepedia.org/annals-of-mathematics), using a virtually abelian group and the field of order two.<sup>[7](https://annals.math.princeton.edu/2021/194-3/p09)</sup> Alan Murray then extended the discovery, presenting a family of counterexamples for every prime characteristic.<sup>[15](https://arxiv.org/abs/2106.02147)</sup> Gardam later showed that the unit conjecture is also false in characteristic zero.<sup>[16](https://arxiv.org/abs/2312.05240)</sup>

**The zero-divisor conjecture.** Kaplansky's zero-divisor conjecture asserts that for K a field and G a torsion-free group, K[G] has no nontrivial zero divisors. Unlike the unit conjecture, it remains open; the search for further counterexamples and for proofs in restricted settings, such as ordered groups, continues.<sup>[17](https://arxiv.org/abs/2501.07646)</sup>

## History and applications

The concept of the group ring is relatively old. It appears implicitly in a paper of Augustus Cayley that is considered the first work in the abstract theory of groups, and it was explicitly introduced by [Theodor Molien](https://www.edgepedia.org/theodor-molien) in 1897. The concept became important through its applications in the theory of group representations, after the work of R. Brauer, F. G. Frobenius, E. Noether, and I. Schur.<sup>[18](https://www.ime.usp.br/~polcino/group_rings/)</sup>

Beyond representation theory, group rings are used in homological algebra, algebraic topology, and algebraic K-theory, and significant applications have been obtained in the theory of error-correcting codes used in digital transmissions, allowing the creation of codes that are simultaneously efficient and reliable.<sup>[18](https://www.ime.usp.br/~polcino/group_rings/)</sup> In topology, the group ring of the fundamental group appears in chain complexes of universal covers, and the algebraic K- and L-theory of group rings, \( K_n(RG) \) and \( L_n(RG) \), is highly significant though hard to compute.<sup>[1](https://alphabetamath.cc/item/def-group-ring)</sup><sup> • </sup><sup>[19](https://www.researchgate.net/publication/45908389_K-_and_L-theory_of_Group_Rings)</sup>

## References

1. [The group ring R[G] of finitely supported formal R-linear combinations of group elements, Alphabeta Math](https://alphabetamath.cc/item/def-group-ring)
2. [Groups of units of orders, University of Luxembourg thesis](https://math.uni.lu/kiefer/Publications/thesis.pdf)
3. [Class notes, Definition 2.2.1 (semigroup ring), Michigan State University](https://users.math.msu.edu/users/meierfra/Classnotes/M818F12/M818F12notes.pdf)
4. [M. Khalkhali, From groups to Hopf algebras and quantum groups, Nagoya University lecture notes](https://www.math.nagoya-u.ac.jp/~richard/teaching/s2015/Khalkhali_13.pdf)
5. [The standard Hopf structure on a group algebra](https://utensil.github.io/forest/fgap-001V/)
6. [Group Representations as Modules over the Group Algebra, KEVOS](https://kevos.com/engineering/group-representations-and-modules)
7. [Giles Gardam (2021). A counterexample to the unit conjecture for group rings. Annals of Mathematics 194(3), 967–979.](https://annals.math.princeton.edu/2021/194-3/p09)
8. [MIT 18.769 Lecture 5 Notes](https://notes.edubirdie.com/docs/massachusetts-institute-of-technology/18-769-topics-in-lie-theory-tensor-ca/86435-18-769-lecture-5-notes)
9. [Hopf Algebras, ScienceDirect Topics](https://www.sciencedirect.com/topics/mathematics/hopf-algebras)
10. [Chapter 6: The group algebra, Tel Aviv University course notes](https://www.math.tau.ac.il/~borovoi/courses/ReprFG/Hatzagot.pdf)
11. [Romyar Sharifi, Abstract Algebra, Chapter 13: Representation theory, UCLA](https://math.ucla.edu/~sharifi/notes/algebra-ch13.html)
12. [Peter Webb, A Course in Finite Group Representation Theory, University of Minnesota](https://www-users.cse.umn.edu/~webb/RepBook/RepBookLatex.pdf)
13. [Structure of group rings and the group of units of integral group rings: an invitation, arXiv:2008.11569](https://arxiv.org/abs/2008.11569)
14. [L. Margolis and Á. del Río (2019). Finite subgroups of group rings: A survey. Advances in Group Theory and Applications 8, 1–37.](https://cris.technion.ac.il/en/publications/finite-subgroups-of-group-rings-a-survey/)
15. [Alan G. Murray (2021). More Counterexamples to the Unit Conjecture for Group Rings. arXiv:2106.02147](https://arxiv.org/abs/2106.02147)
16. [Giles Gardam (2023). Non-trivial units of complex group rings. arXiv:2312.05240](https://arxiv.org/abs/2312.05240)
17. [On zero-divisors and units in group rings of torsion-free CAT(0) groups, arXiv:2501.07646](https://arxiv.org/abs/2501.07646)
18. [Group rings and Noncommutative Rings, Instituto de Matemática e Estatística, Universidade de São Paulo](https://www.ime.usp.br/~polcino/group_rings/)
19. [K- and L-theory of Group Rings, ResearchGate](https://www.researchgate.net/publication/45908389_K-_and_L-theory_of_Group_Rings)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Hopf and quantum algebras › Hopf algebra structure and examples*

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