Ideals in group algebras of simple locally finite groups of 1-type.
This paper continues the work of \textit{A.~E.~Zalesskij} and many others on the ideal lattice of characteristic 0 group algebras \(K[G]\) of infinite locally finite simple groups \(G\) [see Finite and locally finite groups, NATO ASI Ser., Ser. C, Math. Phys. Sci. 471, 219-246 (1995; Zbl 0839.16021)]. It is known that many of these group algebras have only three 2-sided ideals, namely 0, \(K[G]\) itself and the augmentation ideal \(\omega(K[G])\). Indeed, even when this is not the case, it appears that \(K[G]\) has surprisingly few ideals in general. In the PhD thesis of \textit{S.~Delcroix} [Non-finitary locally finite simple groups (Univ. Gent, Belgium) (2000)] the infinite locally finite simple groups were divided into four categories, namely finitary linear groups, groups of 1-type, groups of \(p\)-type, and groups of \(\infty\)-type. The main results here concern groups of 1-type. Specifically, it is shown that if \(G\) is of 1-type, then \(K[G]\) has at least four ideals. Furthermore, if \(G\) is a direct limit of finite direct products of alternating groups, then the ideal lattice of \(K[G]\) consists of a strictly descending chain \(K[G]=I_0\supset I_1=\omega(K[G])\supset I_2\supset\cdots\) with intersection 0. A key ingredient in this work is a result of \textit{U.~Meierfrankenfeld} [in Finite and locally finite groups, NATO ASI Ser., Ser. C, Math. Phys. Sci. 471, 189-212 (1995; Zbl 0858.20026)].
non-finitary locally finite simple groups
scientific article; zbMATH DE number 2101030