Brief history

The concept of Group Ring is relatively old. It appears implicitly in a paper of A. Cayley [8], which is considered the first work in the abstract theory of groups [ 8], and was explicitly introduced by T. Molien in 1897. The concept became important because of its aplications in the theory of group representations, after the works of R. Brauer, F. G. Frobenius, E. Noether and I. Schur (see, for example, [ 5], [ 6] and [49 ]).

In 1940 G. Higman [34] published the first paper directly related with the so-called Isomorphism Problem which, in general, remains open till now. In the sixties the subject began to draw great attention from scientist, and was also stimulated by the inclusion of questions about group rings in the famous list of problems in ring theory by I. Kaplansky [37], [ 38]. Thus, group rings entered the area of interest of ring theorists. The first article in this direction, due to I.G. Connell [ 13], was highly stimulating to the area as well as the inclusion of chapters on group rings in the books by J. Lambeck [ 41] and P. Ribemboim [60]. The almost simultaneous publication of two excelent monographs on the subject with little intersection (see [62] and [ 52]), makes clear the level of the development of the theory, already by the middle of the seventies.


The importance of the subject

The theory of group rings is a meeting point of various algebraic theories. Taking into account its central role in the development of the theory of group representations, it is clear how close the later is related to it. The direct connections with the theory of groups and with ring theory are also obvious because of the intimate relations between the results on group rings and deep facts about groups and rings. Since integral group rings are of special interest for researchers and the topic involves the use of algebraic numbers, algebraic number theory also has an important role in the development of the subject. Finally, it is worthwhile to mention that group rings are important in other branches of mathematics, such as homological algebra, algebraic topology and algebraic K-theory, and that during the last decade significant applications have been obtained in the theory of error correcting codes which are used in digital transmissions, allowing the creation of new codes which are simultaneously efficient and reliable.

To get an idea about the importance of group rings in algebraic research, it is enough to observe that several great contemporary algebraists have worked at some point of their lives in the area, contributing fundamentally to its development. Among them we can mention: S.A. Amitsur, H. Bass, E. Formanek, N.D. Gupta, I.N. Herstein, G. Higman, A.V. Jategaonkar, I. Kaplansky, W. May, K.W. Roggenkamp, W. Rudin e H.J. Zassenhaus. The names of D.S. Passman and S.K. Sehgal have to be added to this list, due to their enormous contribution to the area. Also in the ex-Soviet Union there was great interest in the area of group rings which was represented by such known scientists as S.D. Berman, A.A. Bovdi, A.E. Zalesskii and A.V. Mikhalev.

Nowadays reseach in group rings is rather intense. Consulting the Mathematical Reviews, for example, one discovers that practically each month several articles are published in the area. Moreover, group rings occupy an important place in various international conferences both in group theory and ring theory. For instance, great attention was dedicated to the area on the Kananaskis' meeting on ``Group rings and representation theory'' in February of 1998 in Canada and almost half of the talks in the Trento's conference in Italy on ``Methods in representation theory'', held in May of 1998, were dedicated to the subject. Something similar happened in 1999 in the conference on ``Group rings and Hopf algebras'' held in April in St. John's, Canada. We can also mention the traditional conference in ``Groups and Group Rings'' being held anually in East Europe (mainly in Poland) since 1993.

The Institute of Mathematics of the University of São Paulo has organized two international workshops on ``Group rings and related topics'', one in July of 1995 and another in February of 1998.


Some of the main problems in the area

Let $R$ be a unital ring and $G$ an arbitrary group. We define $RG$ as the set of all linear combinations of the form

\begin{displaymath}\alpha = \sum _{g \in G} a(g) g\end{displaymath}

where $a(g) \in R$ and $a(g) = 0 $ except of a finite number of coefficients. Note that the definition implies that two elements $\alpha = \sum_{g \in G} a(g)g, \beta =\sum_{g \in G} b(g)g \in RG$ are equal if and only if $a(g) = b(g) , \forall g \in G$ .

The sum of elements of $RG$ is defined by:


\begin{displaymath}( \sum_{g \in G} a(g)g ) + ( \sum_{g \in G} b(g)g ) = \sum_{g \in G} ( a(g) + b(g) )g.\end{displaymath}

Also, given two elements $\alpha = \sum_{g \in G} a(g)g, \beta =\sum_{g \in G} b(g)g \in RG$ their product can be defined by:


\begin{displaymath}\alpha \beta = \sum_{g,h \in G} a(g)b(h) gh .\end{displaymath}

It is easily verified that with the above operations $RG$ is a ring, which is called the group ring of $G$ over $K$ .

Moreover, we can define the product of an element from $RG$ by elements of $R$ by putting:


\begin{displaymath}\lambda ( \sum_{g \in G}a(g)g ) = \sum_{g \in G} (\lambda a(g))g.\end{displaymath}

Again, it is easily seen that $RG$ is an $R$ -module. Moreover, if $R$ is commutative, it follows that $RG$ is an algebra over $R$ , and in this case it is frequently called the group algebra of $G$ over $R$ .

The map $\varepsilon :RG \rightarrow R$ given by $\varepsilon (\sum_{g \in G}a(g)g ) = \sum_{g \in G}a(g)$ is a ring homomorphism; it is called the augmentation map of $RG$ and plays a central role in the theory.

In view of the intimate connection with representation theory, it is natural to ask to which extent the knowledge of the structure and properties of a group ring $RG$ determines $G$ . More formally, consider the following question:



Given two groups $G$ and $H$ and a ring $R$ , is it true that the existence of an isomorphism $RG\cong RH$ implies that $G\cong H$ ?

It is easy to show, for example, that two non-isomorphic finite abelian groups of the same order have isomorphic group algebras over the complex numbers. However, in 1950 S. Perlis and C. Walker [ 53 ] proved that finite abelian groups are in fact determined by their group ring over the field of rational numbers. Later in 1956, W.E. Deskins [15] showed that finite abelian $p$ -groups are determined by their group rings over arbitrary fields of characteristic $p$ . Some partial results on non-commutative groups were obtained by D.B. Coleman [11] and D.S. Passman [50 ], [51].

This seemed to suggest that for particular families of groups there should exist appropriate fields for which the isomorphism problem could be solved positively. However, in 1972, E. Dade [ 14 ] published an example of two non-isomorphic finite groups (with relatively simple structure) whose group rings are isomorphic over all fields $K$ !

As a consequence, attention was principally concentrated on integral group rings of finite groups considering the following conjecture:



(ISO)${\rm Z}\!\!\!{\rm Z}G \cong {\rm Z}\!\!\!{\rm Z}H \Longrightarrow G \cong H.$



One of the reasons to turn to the integral case is that ${\rm Z}\!\!\!{\rm Z}G \cong {\rm Z}\!\!\!{\rm Z}H$ implies $RG\cong RH$ for all other commutative rings $R$ . Thus, in this context, the isomorphism ${\rm Z}\!\!\!{\rm Z}G \cong {\rm Z}\!\!\!{\rm Z}H$ provides the strongest possible hypothesis. The first positive results on this conjecture were obtained in 1940 by Graham Higman [ 34] who proved it for finite abelian groups and for hamiltonian $2$ -groups. Untill now the problem has not been solved completely, but several deep results have been obtained for various classes of groups.

We remind that an isomorphism $\varphi :{\rm Z}\!\!\!{\rm Z}G \rightarrow {\rm Z}\!\!\!{\rm Z}H$ is called normalized if, for every element $\alpha \in {\rm Z}\!\!\!{\rm Z}G$ we have that $\varphi (\alpha )=\varepsilon (\varphi (\alpha ))$ (or, equivalently, if $\varepsilon (\varphi (g)) = 1$ for every element $g\in G$ ).

It is easy to show that if there exists an isomorphism $\varphi :{\rm Z}\!\!\!{\rm Z}G \rightarrow {\rm Z}\!\!\!{\rm Z}H$ then there also exists a normalized isomorphism between these rings.

Let $\varphi :{\rm Z}\!\!\!{\rm Z}H \rightarrow {\rm Z}\!\!\!{\rm Z}G$ be a normalized isomorphism. Note that if $\varphi (h) \in G$ , for all $h\in H$ , then $\varphi $ itself, by restriction, gives an isomorphism between $H$ and $G$ . In fact this was the way used by G. Higman to prove the results mentioned above. The big difficulty is to get major knowledge about the elements of the form $\varphi (h)$ , $h\in H$ . Observe that since $\vert H\vert=n$ , we have $h^n=1$ for all $h\in H$ and, since $\varphi $ is a morphism, it follows that $\varphi (h)^n=1$ . This means that $\varphi (h)$ , $h\in H$ , is always a unit of finite order in ${\rm Z}\!\!\!{\rm Z}G$ .

Let $G$ be a finite group. We define:

\begin{displaymath}{\cal U}({\rm Z}\!\!\!{\rm Z}G) = \{\alpha \in {\rm Z}\!\!\!{\rm Z}G \;\vert\; \alpha \mbox{ is invertible} \},\end{displaymath}


\begin{displaymath}{\cal U}_1({\rm Z}\!\!\!{\rm Z}G) = \{ \alpha \in {\cal U}({\rm Z}\!\!\!{\rm Z}G) \;\vert\; \varepsilon (\alpha )=1 \}.\end{displaymath}

The first set is called the unit group of ${\rm Z}\!\!\!{\rm Z}G$ and the second, which is normal in the first one, is the group of normalized units of ${\rm Z}\!\!\!{\rm Z}G$ .

It is natural to ask what information can be obtained about the group of normalized units of ${\rm Z}\!\!\!{\rm Z}G$ keeping in mind that an adequate knowledge of this group can help in the solution of the isomorphism problem.

In the early seventies, H.J. Zassenhaus formulated various conjectures about units and normalized isomorphisms of integral group rings. They are listed below indicating the notation used in [ 63].



In view of the counter examples constructed to (ZC2) it is natural to look for some substitutions or weaker versions to the Zassenhaus conjectures. The following $p$ -version was introduced in [ 16]:

This conjecture states in particular that any Sylow $p$ -subgroup of ${\cal U}_1({\rm Z}\!\!\!{\rm Z}G)$ is rationally conjugate to a $p$ -subgroup of $G$ , thus suggesting a Sylow-like theorem for ${\cal U}_1({\rm Z}\!\!\!{\rm Z}G)$ .



We can mention other important questions in this context. It is natural to ask how a given group $G$ is localized in the unit group of ${\rm Z}\!\!\!{\rm Z}G$ ; more precisely, determine the normalizer of $G$ in ${\cal U}({\rm Z}\!\!\!{\rm Z}G)$ . Obviously, the centre ${\cal Z}({\cal U}({\rm Z}\!\!\!{\rm Z}G))$ and $G$ itself normalize $G$ . The so-called normalizer conjecture says that these two groups determine the entire normalizer. More precisely:


\begin{displaymath}{\cal N}_{{\cal U}({\rm Z}\!\!\!{\rm Z}G))}(G) = G.{\cal Z}({\cal U}({\rm Z}\!\!\!{\rm Z}G)).\end{displaymath}

The results obtained since the seventies show that the unit group of a group ring has a very complicated structure. One of the reasons is that very often the unit group contains a free group of rank 2. One of the open problems is to completely solve the question of existence of free subgroups in unit groups and to determine concrete generators of such subgroups.

There exist various generalizations of the concept of group ring. Some of them were extensively studied such as the ``skew'' group rings, crossed products, semigroup rings, centralizing extentions, Frobenius and quase-Frobenius rings and Hopf algebras.

One of the interesting generalizations is the concept of loop ring which is the non-associative equivalent of group rings. This concept was introduced by H. Bruck in 1944 [ 7 ] as a way of constructing examples of non-associative rings. The subject drew attention when E.G. Goodaire iniciated in 1983 [ 28 ] the study of a class of loop rings which satisfy important identities: the alternative laws. The research on alternative loop rings is rather active now; the basic reference on this is [ 29].

We also would like to mention that recently, motivated by the study of C$^*$ -algebras, R. Exel [ 18] and, independently, J. C. Quigg and I. Raeburn [ 59 ] introduced partial representations of groups, which naturally lead to the definition of partial group rings. These have interesting properties, in particular, with respect to the isomorphism problem. R. Exel observed that unlike in the group ring case, an abelian group of order $4$ is determined up to isomorphisms by its partial group algebra over the complex numbers.

Finally, observe that in the most simple case of a group ring $K G$ of a finite group $G$ over a field $K$ whose characteristic does not divide the group order it is well known that $K G$ is isomorphic to a direct sum of matrix rings with coeficients in division rings. Because of this fact, the investigation of a number of problems in the theory of group rings begins with their study in the case of division rings.

For example, the question of the existence of free subgroups in the unit groups of group rings motivated a similar conjecture for division rings due to A.Lichtman [43 ].

(G) The multiplicative group of a non-commutative division ring contains a free subgroup of rank $2$ .

There exists a similar conjecture due to L. Makar-Limanov [ 47].

(A) A finitely generated division ring of infinite dimension over its centre $K$ , contains a free $K$ -algebra of rank $2$ .

Both conjectures can be combined in the following more ambitious form:

(AG) A finitely generated division ring of infinite dimention over its centre $K$ contains the group algebra over $K$ of the free group of rank $2$ .


The actual stage of knowledge

We start with the isomorphism problem. A counter example to the problem was recently announced by M. Hertweck (see [ 39 ]), but it is not yet published. Even if the announcement is confirmed, it is still an important problem to decide for which classes of groups the conjecture holds. It has been proved in the following cases:

The Zassenhaus conjectures drew great attention since their formulation and continue to be intensively investigated till now.

We give below the list of groups for which (ZC1) has been verified.

The conjecture (ZC3) was proved for the following groups.

On the other hand, the validity of the conjecture (p-ZC) was established for the folowing families of groups:

The first three results have been proved by M. Dokuchaev and S.O. Juriaans [16] and the last one by M. Dokuchaev, S.O. Juriaans and C. Polcino Milies [ 17].

As to the normalizer conjecture we remind that it was proved in 1964 by D.B. Coleman [ 12] for finite p-groups. Later S. Jackowski and Z. Marciniak showed in 1987 that it holds for finite groups with normal Sylow $2$ -subgroup (and, in particular, for groups of odd order) [ 23 ]. A very recent progress is due to Y. Li, M.M. Parmenter and S.K. Sehgal who proved that the conjecture also holds for finite groups whose non-normal subgroups have a non-trivial intersection [ 42].

The existence of free subgroups in the unit groups of group rings of finite groups was established in [ 33] by B. Hartley and P.F. Pickel over the integers and by J.Z. Gonçalves over fields in [21 ] and [22]. Recently, Z. Marciniak and S.K. Sehgal [ 48], J.Z. Gonçalves and D.S. Passman [ 25] and J.Z. Gonçalves, A. Mandel and M. Shirvani [ 24] showed how to generate free groups using well known units [ 48].



With respect to alternative loop rings a number of important results were obtained in the nineties, such as the description of RA loops, i.e., the loops whose loop algebras over every ring of characteristic different from $2$ is alternative (O. Chein and E.G. Goodaire [ 9]), the validity of the isomorphism problem over the integers and (Aut) (E.G. Goodaire and C. Polcino Milies [ 30]), the validity of the Zassenhaus conjectures (E.G. Goodaire and C. Polcino Milies [ 31], [32]), the classification of finite indecomposable RA loops (E. Jespers, G. Leal e C. Polcino Milies [ 36]), etc.

Recently a book was published [ 29] which contains a systematic account of the actual knowledge on the subject.

Considering the most ambitious conjecture about units in division rings, namely (AG), it is to be mentioned that recent positive results by L.M.V. Figueiredo, J.Z. Gonçalves and M. Shirvani [ 20] and de J.Z. Gonçalves and M. Shirvani [ 26] and [27 ] permit to believe in its correctness.


The Research Project

The research on group rings at IME-USP has a long history of colaboration of our professors among themselves and with a number of scientists from universities either in Brazil or abroad. The aim of the present project is to continue this collaboration, developing it in various directions as well as broadening it by establishing new scientific contacts.



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