Group Algebras: Normal Subgroups and Ideals

The standard correspondence between the normal subgroups of the group G and some ideals of the group algebra FG is described. There is the problem of what we can say (or even prove) about a two-sided ideal \(I \neq (0)\) of \({\mathbb{F}}G\) that does not contain any element of the form 1 − g ≠ 0, g ∈G of the standard basis of the augmentation ideal of \({\mathbb{F}}G\) . The main part of the argument of [2] yields the insight that, for such an ideal I there exists an expansion \(H \supseteq G\) such that the ideal J of \({\mathbb{F}}H\) spanned by I contains an element 1 − h, h ∈ H \ G. Using the ideas of [2], we construct \({\mathbb{F}}\) -thick groups H such that for every ideal J ≠ (0) of \({\mathbb{F}}H\) there are elements 1 − h ≠ 0 in J. This construction allows many variations. Examples of simple \({\mathbb{F}}\) -thick groups were pointed out in [2]. A natural class of (in general non-simple) \({\mathbb{F}}\) -full groups are the normal sections of the groups $$S^{\prime} := \rm{Sym}({\it M})/ \rm{Fin}({\it M})\,\rm{for\,any\,infinite\,set} {\it M}.$$ (Here, Fin(M) is the subgroup of all finitary permutations of M.)