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Large hereditarily just infinite groups. (English) Zbl 1209.20028
The author gives a criterion for an inverse limit of a sequence of finite groups to be hereditarily just infinite, and a similar criterion for direct limits. He then uses this criterion to give constructions of hereditarily just infinite groups which satisfy required properties. In particular, the author constructs a hereditarily just infinite profinite group in which every countably based profinite group can be embedded (Theorem A). Thus demonstrating that a hereditarily just infinite profinite group need not be virtually pro-\(p\). Secondly the author constructs hereditarily just infinite prosoluble groups that are not (topologically) finitely generated (Theorem B).
This is a useful paper which provides examples of hereditarily just infinite groups which enhance our understanding of these structures.

MSC:
20E18 Limits, profinite groups
20E07 Subgroup theorems; subgroup growth
20E26 Residual properties and generalizations; residually finite groups
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