Group Algebra KG Research Articles

Some self-dual local rings of integers not free over their associated orders

Let p be a prime number, and let K be a finite extension of the rational p-adic field ℚp. Let L/K be a finite abelian extension with Galois group G, and let L, K denote the valuation rings of L, K respectively. Then L is a free module of rank 1 over the group algebra KG. Defining the associated order of the extension L/K byL can be viewed as a module over the ring , and a fortiori over the group ring KG.

Lie Dimension Subgroups, Lie Nilpotency Indices, and the Exponent of the Group of Normalized Units

Let K be a field of characteristic p > 0, and let G be a finite p-group. Let U be the group of normalized units of the modular group algebra KG. In this paper we study the relation between exp (U) and exp (G). The main result shows that, if p ⩾ 7 and exp(G)3 > |G|, then G and U have the same exponent. We also show that, in general, exp(U) cannot be bounded above by any fixed function of exp(G). The method involves a reduction to problems in Lie nilpotency indices, which are solved via an extensive study of Lie dimension subgroups. Some results for smaller p are also given.

A remark on the Loewy-layers of the modular group algebra of a finite p-group

We consider the dimensions c i of the Loewy factors of the modular group algebra KG of a finite p-group G. It is well-known that c i = c s−i where s is the Loewy length of KG. For p = 3 and 5 we describe p-groups with the property that c i ⩽ c i−1 for some i⩽ 1 2 S thus settling an old open problem. Also, the similar question for restricted Lie algebras with nilpotent p-map is studied.

On modules with cyclic vertices in the Auslander-Reiten quiver

In studying indecomposable modules of a finite-dimensional K-algebra where K is a field, the Auslander-Reiten quiver r has proved to be a powerful tool. If the algebra is a group algebra KG (or a block B), then properties of the group are related to the graph structure of the Auslander -Reiten quiver. For example, the graph f(KG) (or T(B)) is finite if and only if a Sylow-p-subgroup of G (or a defect group of B) is cyclic, where p is the characteristic of K. It is now well known that an indecomposable KG-module has a vertex, which is a minimal subgroup Q of G such that A4 is Q-projective; and Q is unique up to G-conjugation [7]. We ask whether there are constraints on the position of modules with a given vertex in the Auslander-Reiten quiver. Here we are concerned with this problem for modules with cyclic vertices which are not projective. There is not much to say in the case when the defect group of the block is cyclic; in this case, all its modules have cyclic vertices. We assume therefore that the block containing these modules has a non-cyclic defect group. Assume that M is indecomposable with a cyclic vertex, and that A4 is not projective. It is well known that M is n-periodic, hence r-periodic. (For group algebras, the Auslander-Reiten translate r is the same as 02, where 52 is the usual Heller operator.) By a more general theorem [2, p. 1631, the component--O say-of the stable Auslander-Reiten quiver containing M, is a “tube” [ 151. Given such a tube 0, then the vertices in each row form a single r-orbit. The “end” of the tube is the unique row where each vertex has only one predecessor. That is, if M is a module corresponding to a vertex [it41 in 0 and O+rM-+E+M-+O (*I 289 002 l-8693/86 $3.00

On the nilpotency index of the radical of a group algebra, VII

Let p be a fixed prime number, let K be a field of characteristic p, let G be a finite p-solvable group with a p-Sylow subgroup P of order pm, and let t(G) be the nilpotency index of the radical J(KG) of a group algebra KG of G over K. Moreover we set M= O,.(G), H= O,.,,(G), pr = 1 H/MI, and F/M a Frattini subgroup of H/M. Then G/H is isomorphic to a subgroup of GL(H/F) where H/F is regarded as a vector space over GF(p) (see [2, Lemma 1.2.53). This assertion shall be used freely in this paper without references. Every subgroup of GL(2, 3) acts naturally on the elementary abelian group E of order 9. Let T and S be a semidirect product of E by GL(2, 3) and SL(2, 3) with respect to this action,. respectively. Y. Tsushima [13] proved the inequality p” 2 t(G). In the light of this inequality he [ 143 (see also [ 1 l] ) proved that t(G) = pm if and only if P is cyclic. Further S. Koshitani [S, 61 (see also [ 111) has proved the following conditions are equivalent for p 2 3 and m 2 2:

Representation groups for the Schur index

Fix an algebraic number field K and an irreducible (complex) character x of some finite group G. Assume K = K(X) contains the values of x so that the simple component A = .4(x, K) of the group algebra KG determined by 1 is central over K. It is a basic consequence of the Brauer-Witt theorem that A is a so-called cyclotomic algebra (see [lS]). We exhibit here a slightly different point of view, namely that of constructing a “representation group” for A, similar to the concept used in Clifford theory. From class field theory it is known that the order of the class [A] of A in the Brauer group Br(K) equals its index m,(x). It suffices to study the primary components of [A]. So fix a prime p and let [A], denote the pcomponent of [A] in Br(K). Let K, be the field obtained from K by adjoining the nth roots of unity. If n = exp(G) then by Brauer K, is a splitting field for A (and G). We shall see that there is a field L s K, with p-power degree over K such that

On linear groups over a field of fractions of a polycyclic group ring

LetG be a torsion free polycyclic-by-finite group andD be the field of fractions of the group algebraKG. Then any periodic subgroup ofDn is locally finite. This answers a question posed by D. Farkas.

Group algebras of finite p-solvable groups with radicals of the fourth power zero

SynopsisLet J(KG) be the Jacobson radical of the group algebra KG of a finite p-solvable group G over a field K of characteristic p > 0, and let t(G) be the least positive integer t such that J(KG)t = 0. In this paper we determine the structure of G with t(G) = 4 under the assumption that H is abelian, H is metacyclic, or the order of H is not divisible by 3 where H = O2'(G).

Gyclotomic Division Algebras

Let K be a field of characteristic zero. The Schur subgroup S(K) of Brauer group B(K) consists of those equivalence classes [A] which contain an algebra which is isomorphic to a simple summand of the group algebra KG for some finite group G. It is well known that the classes in S(K) are represented by cyclotomic algebras, (see [16]). However it is not necessarily the case that the division algebra representatives of these classes are themselves cyclotomic. The main result of this paper is to provide necessary and sufficient conditions for the latter to occur when K is any algebraic number field.Next we provide necessary and sufficient conditions for the Schur group of a local field to be induced from the Schur group of an arbitrary subfield. We obtain a corollary from this result which links it to the main result. Finally we link the concept of the stufe of a number field to the existence of certain quaternion division algebras in S(K).

Group rings of hyperabelian groups

Ifk is a field of characteristicp>0 then we find all hyper-(Abelian or locally finite-p′) groupsG such that the augmentation ideal of the group algebrakG has the AR property.

Inducedp-elements in the Schur group

The main result of this paper gives necessary and sufficient conditions for the ^-primary part S(K)P of the Schur group S (K) to be induced from S (F)p for any subfield F of K where K is contained in Q (en), under the restriction that ep2 is not in K if p > 2 and n is odd if p~2, where en is a primitive nth root of unity. Moreover we completely answer the question: When is SiQien + e-1)) induced from S(Q)? for any n, and also the question: When are the quaternion division algebras in S(Q(en)) induced from S(Q(en -f e; 1))? for any n. Finally, in the last section we investigate the generalized group of algebras with uniformly distributed invariants which we introduced in an earlier paper. We obtain, for the first time, a sufficient condition for the group to be induced from a certain subgroup. Preliminaries* Let L = Q(en) and let K be a subfield of L. Although it is not necessary for all results in the paper it is convenient to choose n as small as possible for a given K. The Schur subgroup S(K) of the Brauer group B(K) consists of those equivalence classes [A] which contain an algebra which is isomorphic to a simple summand of the group algebra KG for some finite group G. An elegant proof of the following result was given by Janusz [18, Prop. 6.2, p. 89]:

More on the AR property and chain conditions in group rings

We investigate when the augmentation ideal of a group algebrakG has the AR property in the cases ofG a locally nilpotent or hyperabelian group.

Group algebras of periodic groups of a solvable multiplicative group

The structure of the group algebra KG of an arbitrary periodic group G such that the multiplicative group U(KG) is solvable is completely determined.

Normal complements in modp-envelopes

SupposeG is a finitep-group andk is the field ofp elements, and letU be the augmentation ideal of the group algebrakG. We investigate whichp-groups,G, have normal complements in their modp-envelope,G*.G* is defined byG*={1−u∶u∈U}.

Maximal quotient rings of prime group algebras

It is shown that a prime nonsingular group algebra KG, where G is a group whose conjugacy classes are countable, has for its maximal right quotient ring either a full linear ring or a simple directly infinite ring. In the case where G is also locally finite only the second type can occur.

A deformation-theoretic version of Maschke's theorem for modular group algebras: The commutative case

Let G be a finite group and k a field of characteristic p > 0. Maschke’s theorem says that the group algebra KG is semisimple if and only if p does not divide the group order, ) G I. We refer to RG as a modular group algebra when p divides / G I. In this case the representation theory of KG presents many unsolved problems. Of special interest is the structure of the indecomposable modules for kG, even when G is an abelian non-cyclic p-group [ 1, p. 431; 7, Introduction]. In this paper we begin a study of modular group algebras and their representations using the deformation theory of Gerstenhaber [4] and NijenhuisRichardson [6]. A major intuition in the program derives from the correlation for k-algebras between deformability and the failure of semi-simplicity. In Section 1 we show how, by using deformations, Maschke’s theorem may be recovered for abelian modular group algebras. One might conjecture furthermore that any modular group algebra deforms into a predictable semisimple algebra. Some non-abelian examples in Section 2 support this conjecture. There we exhibit deformations of certain group algebras of the dihedral and symmetric groups into the “expected” semisimple algebras. In Section 3 we propose further questions and avenues for investigation.

Passman-Zalesskii Radical of group algebras

Recently Passman (attributing the origin of the idea to Zalesskii) has defined the following ideal in a ring, (2).Definition. For a unitary ring R,N * R = {α ∈ R | αS is nilpotent for all finitely generated subrings S of R}.For a group algebra KG over a field K of characteristic p ≠ 0, he has proved the radical property:

Nilpotency of the multiplicative group of a group ring

It is proven that if K is a commutative ring of characteristic pm while group G contains p-elements, then the multiplicative group UKG of group ring KG is nilpotent if and only if G is nilpotent and its commutant G′ is a finite p-group. Those group algebras KG are described for which the nilpotency classes of groups G and UKG coincide.

On the semisimplicity of modular group algebras

Let G be a discrete group, let K be a field and let KG denote the group algebra of G over K. We say that KG is semisimple if its Jacobson radical JKG is zero. If K has characteristic 0 and K is not an algebraic extension of the rationals then it is known [l, Theorem 1 ] that KG is semisimple. Moreover it appears likely that in the remaining characteristic 0 cases we also have semisimplicity. Thus nothing particularly interesting occurs here. If K has characteristic p>0 and G is a p'-group (that is, G has no elements of order p), then it is known (see [2]) that KG has no nil ideals and that for suitably big fields the group algebra KG is semisimple. Again it appears that for the remaining fields we also have semisimplicity. The interest in characteristic p stems from the fact that, unlike the case in which G is finite, it is quite possible for G to have elements of order p and yet have the group algebra KG be semisimple. Several examples of such groups have been exhibited and in each case a big abelian group is involved as either a subgroup or a factor group. In this paper we study the group algebras KG of those groups G having a big abelian subgroup or factor group. The methods used are extremely elementary. Two interesting examples of the type of groups which we can deal with are as follows. Let P be a cyclic group of order p and let C be an infinite cyclic group. Let Gi = C P and G2=P I C where I denotes the restricted Wreath product. Now Gi has a normal torsion free abelian subgroup of finite index p and G2 has a normal elementary abelian p-subgroup E with G2/E infinite cyclic. Surprising as it may seem if K is an algebraically closed field of characteristic p, then 7<Gi and KG2 are both semisimple. For the remainder of this paper K is an algebraically closed field of characteristic p. By a linear 7I-character of KG we mean a FJ-homomorphism X: KG—*K.

On algebras with a finite number of indecomposable modules

Let A be an associative algebra over a field K such that the dimension, [A: K], of A over K is finite. Following the terminology of [8], we shall say that A is of bounded module type if there exists an integer n such that for every indecomposable left A-module M, the inequality [M: K] ? n holds. The algebra A is of finite module type if A has only a finite number of nonisomorphic indecomposable left modules. Clearly, algebras of finite module type are of bounded module type. The converse to this statement has been conjectured for some time [8], and it appears to be quite difficult to prove. There are some classes of algebras (more general than semi-simple algebras) which are known to be of finite module type. Higman has shown [7] that the group algebra KG of a finite group G over a field K of characteristic p is of finite module type if and only if the p-Sylow groups of G are cyclic. Nakayama [13] introduced the class of generalized uniserial algebras, and showed that they are all of finite module type. It has been known for some time, however, that these two classes do not contain all algebras of finite module type [16]. Before stating the main theorem of this paper, we introduce some notations and definitions. We shall be concerned only with finite dimensional algebras. We shall denote the radical of an algebra A by N. By the socle S(M) of a left A-module M, we mean the sum of the simple submodules of M; S(M) can also be characterized as the set of all elements m in M such that Nm = 0. The socle S(M) of M is the maximal semi-simple submodule of M, and is a direct sum of simple modules. In general, a semisimple module will be called square-free if it is a direct sum of simple modules, no two of which are isomorphic. Note that in a square-free semisimple module M, two simple submodules are isomorphic if and only if they coincide. We can now state the main theorem of this paper.