On Just Infinite Abstract and Profinite Groups

Abstract

An infinite group G is called just infinite if all non-trivial normal subgroups have finite index; if G is profinite it is merely required that all non-trivial closed normal subgroups have finite index. Just infinite groups have arisen in a variety of contexts. The abstract just infinite groups having non-trivial abelian normal subgroups are precisely the space groups whose point groups act rationally irreducibly on the abelian normal subgroups (see McCarthy [7]). Many arithmetic groups are known to be just infinite modulo their centres; examples are SLn (R) for n ≥ 3 and Sp2n (R) for n ≥ 2, where R is the ring of integers of an algebraic number field (see [1]). The Nottingham group over \(\mathop \mathbb{F}\nolimits_p\) (described in Chapter 6) is a just infinite pro-p Groups. Grigorchuk [3], [4] and Gupta and Sidki [5] have introduced and studied some infinite finitely generated p-groups which act on trees, and many, together with their pro-p completions, are just infinite. Using Zorn’s Lemma it is easy to see that if S is either an infinite finitely generated abstract group or an infinite finitely generated pro-p group, then S has a just infinite quotient group. Therefore to decide whether a group-theoretic property implies finiteness, it is sometimes sufficient to consider just infinite groups.KeywordsNormal SubgroupWreath ProductFinite IndexMaximal ConditionBoolean LatticeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.