Group algebras of infinite groups over arbitrary fields

Dunlop, Lilian M (1966) Group algebras of infinite groups over arbitrary fields. MSc(R) thesis, University of Glasgow.

Abstract

In this dissertation, we give an account of some recent work relating to group algebras. In § 2, we define the lover and upper nil radicalo end the Jacobson radical for the group algebra of any group over an arbitrary field and note that for a finite group these radicals coincide. In fact the radical of the group algebra of a finite group over a field is the zero ideal if (1) the field has characteristic zero or (ii) the field has characteristic p (≠ 0) and the group contains no p-elements. In §3, we show that for any algebra with an identity element, over a field whoso cardinal number exceeds the dimension of the algebra over the field, the Jacobson and upper nil radicals coincide (1). These two radicals again coincide for any finitely generated algebra satisfying a polynomial identity (3). These results are used in conjunction with results on the upper nil radical of a group algebra in § 5. Passman (6) has proved that the upper nil radical of the group algebra of any group over a field of characteristic zero is the zero ideal and that if the field has characteristic p ≠ 0, then the group algebra is semi-simple provided that the group contains no p-elements. The main aim of the dissertation is to find condition on the group or the field under which the Jacobson radical of a group algebra is the zero ideal. In § 4, we examine the behaviour of the Jacobson radical of an algebra over a field under extension of the field and establish two theorems by Amitsur on this subject (2). Finally in § 6, using the results established in §§ 3-5, we establish that if the field over which the group algebra is formed is a non-algebraic extension of Q, the field of rational numbers, then the group algebra is semi-simple, whatever the group (4 and 6). We also prove two theorems by Passman (6) on group algebras over fields of characteristic p, in which he shows that if the field is a separably generated , non-algebraic extension of some subfield, or if it is non-denumerable, then the group algebra of any group with no p-elements is semi-simple, Connell (5) has studied the slightly different problem of finding groups which give rise to semi-simple group algebras over arbitrary fields. If the group has no p-elements when the field has non-zero characteristic p then locally finite groups, ordered groups and abelian groups, are such groups, Further, it can be shown that if two groups have semi-simple group algebras over a particular field, then the group algebra of the direct product of the groups over the same field is semi-simple, and that the group algebra of the direct product of any group with the infinite cyclic group over the field is also semi-simple, provided that if the field has non-zero characteristic p, then the group has no p-elements.

• Item Type: Thesis (MSc(R))
• Qualification Level: Masters
• Additional Information: Adviser: D AR Wallace
• Keywords: Mathematics
• Date of Award: 1966
• Depositing User: Enlighten Team
• Unique ID: glathesis:1966-72249
• Copyright: Copyright of this thesis is held by the author.
• Date Deposited: 24 May 2019 15:12
• Last Modified: 24 May 2019 15:12
• URI: https://theses.gla.ac.uk/id/eprint/72249