1940/01/01 by Graham Higman · 7 citations
Mathematics · Computer Science · #Advanced Topics in Algebra #Rings, Modules, and Algebras #Advanced Graph Theory Research #Citation #Group (periodic table) #Library science #Computer science #Mathematics #Combinatorics #Physics
paper · doi:10.1112/plms/s2-46.1.231
openalex publication_date 1940/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/07
If G is any group, written multiplicatively, and K is any ring, then the finite formal sums<br><br>k, g,+ k2' g2 + . . . + Kt Gt, Ki Ɛ K, Gi Ɛ G, (i=lj...,T)<br><br>form a ring, when addition and multiplication is defined inthe obvious way. This ring we call the group ring of Gover K, and we write it as R(G,K). Lore precisely, R(G,K)may be defined as a ring which is a linear set over K, andhas a basis which is a multiplicative group isomorphic to G.If K has the unity element 1, and if go is the identity ofGI then 1.go is a unity element of R(G,K). We shall identify,when confusion cannot arise thereby, elements k of K. withthe elements of k.go of R(G,K). The object of this thesisis to establish certain theorems on the units of R(G,K).<br><br>An element e1 in a ring with unity element 1 will becalled a left unit if there exists also an element e2 suchthat ele2 = 1; and e2 will then be called a right unit, anda right inverse of el. If el is also a right unit, thenit has a uniquely defined inverse which we write as e1-1'and el is then called a unit, simply. We shall deal alwayswith rings R(G,K) in which the coefficient ring K has noright units which are not left units. Whether it is even so possible for R(G,K) to have right units which are notleft units, I do not know. Certainly it is not in themost important cases, - for instance if K can be embeddedin a field, and G is of finite order. Any group ring hasunits of the form e.g, where e is a unit in K, and g is anelement of G, for e.g has the inverse g-1. Such aunit we shall call trivial. One of the questions thatnaturally arises is, for what group rings these are theonly units.<br><br>This thesis is chiefly concerned with the case in whichG is a group of finite order, and K is the ring C of integersin an algebraic field k. We shall then speak of R(G,C) asan integral group ring. R(G,C) is then an order, but notin general a maximal order, of the linear associative algebraR(G,k). Accordingly, after an introductory section, and°heave on definitions, we discuss in Section 3 the group algebraR(G,k). This section is an exposition of well-known factsconcerning the decomposition of the semi-simple algebraR(G,k) into simple components, and is based on the classicaltheory of representations of finite groups in algebraicfields, as developed by I. Schur.<br><br>In Section 4 we pass on to the consideration of integralgrou ings, and more particularly, to a determination of whatsuch rings have only trivial units. The result is, thatR(G,C), where C is the integer ring of the field k, has onlytrivial units in the following four cases, and those only:<br><br>(i) G is Abelian, and the orders of its elements alldivide two; k is the rational field or an imaginary quadraticextension of it;<br><br>(ii) G is Abelian, and the orders of its elements alldivide six; k is the rational field, or the extension of itby a complex cube root of unity;<br><br>(iii) G is Abelian, and the orders of its elements alldivide four; k is the rational field, or the extension of itby a complex fourth root of unity (that is, by i = -1);<br><br>(iv) G is the direct product of a quaternion group andan Abelian group the orders of whose elements all divide two;k is the rational field.<br><br>To establish this theorem, we prove first a theorem which isa particular case of a result due to 0. Schilling, namely,that if k is of finite degree over the rational field theunit group of R(G,C) has a finite index in the unit groupof any maximal order of R(G,k) containing it. Secondly,we show that the group of trivial units in R(G,C) is notcontained as a proper sub-group of finite index in any groupof units of R(G,C). From these two theorems together, itfollows that R(G,C) has only trivial units if and only if amaximal order of R(G,k) containing it has a finite unitgroup. This imposes severe restrictions on the possiblesimple components of R(G,k), which give rise to our mainresult.<br><br>In the following section, we turn our attention to theunits of finite order in R(G,C). '4e find it convenient totreat normalised units only, - that is to say, units inwhich the sum of the coefficients is unity. This involvesno essential loss, since any unit of finite order is theproduct of a normalised unit of finite order and a root ofunity in C; and any group of units of finite order is isomorphicto the direct product of the corresponding group ofnormalised units and a group of roots of unity in C. Interms of normalised units, a theorem from section 4 becomes:The group G is not contained as a proper sub—group of finiteindex in any group of normalised units of ROIG,C). Inparticular this implies that a unit of finite order in thecentrum of R(G,C) must be trivial. If G is Abelian,therefore, all the units of finite order in R(G,C) are.trivial. This is of course no longer true if G is notAbelian. In fact, a unit of finite order in R(G,C) neednot necessarily even be conjugate to a trivial unit. Weare able, however, to prove two results on units of finiteorder. The first states that the elements of any group ofnormalised units of finite order in R(G,C) are linearlyindependent, and even linearly independent module m, forany integer m, and that therefore the order of such a groupcannot exceed the order of G. The second states that theprime factors of the order of a normalised unit of R(G,C) divide the order of G. If we add the assumption that Gis soluble, we can show that the order of the unit dividesthe order of G. Lastly, we show for the very special classof groups whose lower central series terminates in theidentity and whose second derived group consists of theidentity, that a group of normalised units of finite orderin R(G,C) is isomorphic to a subgroup of G. This impliesthat R(G,C) is not isomorphic to any other group ring R(H,C)unless G is isomorphic to H. These last theorems areproved by methods different from those of the rest of thesection, and our chief tools are the two-sided ideals X1.5where H is a self conjugate subgroup of G, generated by theelements h-1, for all h in H. It should be added, thatthroughout this section the coefficient ring C is an arbitraryring of algebraic integers, and may be taken to bering of all algebraic integers.<br><br>In section 6, we apply the theorems we have proved tothe detailed investigation of the ring R(G,C) where G isgenerated by two elements a, b subject to the relations:-<br><br>ap=bp-1=1, b-1ab=av<br><br>where p is an odd prime, and r a primitive number modulo p.Here, too, we show that a group of normalised units inR(G,C) is isomorphic to a subgroup of G.<br><br>Finally, in section 7, we consider group rings ofgroups without elements of finite order. Naturally, thetheorems proved are of an entirely different character.Notably, they do not depend at all on the coefficient ringK, provided that it has no zero divisors. We show, infact, that if G satisfy the condition that every subgroupgenerated by a finite number of elements of G has a homorphismon the free cyclic group, then if K has no zero divisorsneither has R(G,K), and the units of R(G,K) are all trivial.The condition is satisfied by free groups and by freeAbelian groups; and generally, by the direct product andthe free product of any two groups that satisfy it.<br><br>As we have said, section 3 is a repetition of well-knownfacts; and the first theorems of section 4 are a particularcase of a theorem due to Schilling. The rest of the thesisis original, though some of it has been published previously.