Chia Hsin Liu

GROUP ALGEBRAS WITH UNITS SATISFYING A GROUP IDENTITY

Let K[G] be the group algebra of a group G over a field K, and let U(K[G]) be its group of units. A conjecture by Brian Hartley asserts that if G is a torsion group and U(K[G]) satisfies a group identity, then K[G] satisfies a polynomial identity. This was verified earlier in case K is an infinite field. Here we modify the original proof so that it handles fields of all sizes.

Group identities on units of locally finite algebras and twisted group algebras

We study locally finite algebras and twisted group algebras with units satisfying a group identity. As a preliminary result, we obtain a necessary condition for twisted group algebras to satisfy a generalized polynomial identity.

MULTIPLICATIVE JORDAN DECOMPOSITION IN GROUP RINGS OF 2, 3-GROUPS

In this paper, we essentially finish the classification of those finite 2, 3-groups G having integral group rings with the multiplicative Jordan decomposition (MJD) property. If G is abelian or a Hamiltonian 2-group, then it is clear that ℤ[G] satisfies MJD. Thus, we need only consider the nonabelian case. Recall that the 2-groups with MJD were completely determined by Hales, Passi and Wilson, while the corresponding 3-groups were almost completely determined by the present authors. Thus, we are concerned here, for the most part, with groups whose order is divisible by 6. As it turns out, there are precisely three nonabelian 2, 3-groups, of order divisible by 6, with ℤ[G] satisfying MJD. These have orders 6, 12, and 24. In view of another result of Hales, Passi and Wilson, this completes a significant portion of the classification of all finite groups with MJD.

Artinian Hopf algebras are finite dimensional

We prove that an artinian Hopf algebra over a field is finite dimensional. This answers a question of Bergen.

Group Identities and Prime Rings Generated by Units

Abstract Let R be a prime ring and U(R) the group of units of R. We prove that if U(R) generates R and satisfies a group identity,then R is either a domain or a full matrix ring over a finite field.

On nil subsemigroups of rings with group identities

ABSTRACT Let R be a unital ring satisfying a group identity. We prove that if B is a nil subsemigroup of R, then it is locally nilpotent, and is contained in the sum of all nilpotent ideals of R, where the positive integer d is determined by the group identity. Note that the above result for PI-rings is due to Amitsur.

Multiplicative Jordan Decomposition in Group Rings of 3-Groups, II

In this paper, we complete the classification of those finite 3-groups G whose integral group rings have the multiplicative Jordan decomposition property. If G is abelian, then it is clear that ℤ[G] satisfies the multiplicative Jordan decomposition (MJD). In the nonabelian case, we show that ℤ[G] satisfies MJD if and only if G is one of the two nonabelian groups of order 33 = 27.

Groups with Certain Normality Conditions

We classify two types of finite groups with certain normality conditions, namely SSN groups and groups with all noncyclic subgroups normal. These conditions are key ingredients in the study of the multiplicative Jordan decomposition problem for group rings.