Modular group algebra

Let $ F $ be a field and $ G $ a group. The group algebra $ FG $ is called modular if the characteristic of $ F $ is prime, say $ p $, and $ G $ contains an element of order $ p $; otherwise $ FG $ is said to be non-modular.

Practically every result about group algebras over a field of characteristic zero has an appropriate analogue for any non-modular group algebra. Absence of finite skew-fields makes it possible to state certain prime-characteristic analogues in a stronger form. For example, if $ G $ is a finite group, then a non-modular group algebra $ FG $ in characteristic $ p > 0 $ is a sum of matrix algebras over fields (rather than over skew-fields, as in the case of characteristic zero; cf. also Matrix algebra). The theory of modular group algebras is of a much higher level of complexity than that of the non-modular group algebras. For finite $ G $, the topic essentially belongs to the theory of modular representations of finite groups (cf. Finite group, representation of a), and includes such rich pieces as the theory of blocks (cf. Block), Brauer correspondences, the theory of projective and relatively projective modules (cf. Projective module), etc. The radical theory of modular group algebras for $ G $ finite is not very well developed (1996). E.g., there are only fragmentary results about the dimension and the nilpotency index of the radical.

However, if $ G $ is infinite, the most studied case is precisely the theory of radicals, mainly the Jacobson radical. The results obtained centre around two conjectures. The first conjecture, proved for locally solvable groups (cf. Solvable group), states that $ JFG $, the Jacobson radical of $ FG $, coincides with the so-called $ N ^ {*} $- radical, where $ N ^ {*} FG $ is defined as the set of all $ x \in FG $ such that $ xFH $ is nilpotent (cf. Nilpotent group) for every finitely generated subgroup $ H \subset G $. Set

$$ \Lambda ^ {+} = \{ {g \in G } : \left | {H: C _ {H} ( g ) } \right | < \infty $$

$$ \ {} \textrm{ for all finitely generated subgroups } H \subset G \} $$

and let $ \Lambda ^ {p} ( G ) $ be the subgroup of $ \Lambda ^ {+} $ generated by the $ p $- elements. Then $ \Lambda ^ {+} $ is locally finite (i.e., every finite set is contained in a finite subgroup) and $ N ^ {*} FG $ is generated by the Jacobson radical of $ F \Lambda ^ {p} ( G ) $. This reduces the problem to locally finite groups.

The second conjecture concerns locally finite groups and has recently been settled affirmatively by D.S. Passman [a2]. To state it, let $ O _ {p} ( G ) $ be the maximal normal $ p $- subgroup of the locally finite group $ G $ and let $ S ( G ) $ be the subgroup generated by all finite subgroups $ A \subset G $ such that $ A $ is generated by $ p $- elements and $ A $ is a subnormal subgroup in every finite subgroup $ B \subset G $ with $ A \subset B $. Let $ T ( G ) $ be the pre-image of $ S ( G/O _ {p} ( G ) ) $ in $ G $. Then $ JFG $ is generated (as an ideal) by $ JFT ( G ) $, and $ JFT ( G ) $ is easily described in terms of radicals of the group rings of finite groups. Thus, the problem is in fact reduced to the case of finite groups, which is usually regarded as a satisfactory solution for a problem in the theory of infinite groups.

For a locally finite $ G $, the two-sided ideals $ I $ of $ FG $ such that $ J ( FG/I ) = 0 $ can be described in terms of the representations of finite subgroups of $ G $, [a3]. This also provides an efficient machinery for studying the lattice of two-sided ideals of $ FG $.

References

[a1]	D.S. Passman, "The algebraic structure of group rings" , Wiley (1977)
[a2]	D.S. Passman, "The Jacobson radical of group rings of locally finite groups" Adv. in Math. (to appear)
[a3]	A.E. Zalessskii, "Group rings of simple locally finite groups" , Finite and Locally Finite Groups , Kluwer Acad. Publ. (1995) pp. 219–246