Wilson, John S. Large hereditarily just infinite groups. (English) Zbl 1209.20028 J. Algebra 324, No. 2, 248-255 (2010). 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. Reviewer: Rachel D. Camina (Cambridge) Cited in 2 ReviewsCited in 7 Documents MSC: 20E18 Limits, profinite groups 20E07 Subgroup theorems; subgroup growth 20E26 Residual properties and generalizations; residually finite groups Keywords:just infinite groups; profinite groups; pro-\(p\)-groups; subgroups of finite index; inverse limits of finite groups; direct limits; prosoluble groups PDF BibTeX XML Cite \textit{J. S. Wilson}, J. Algebra 324, No. 2, 248--255 (2010; Zbl 1209.20028) Full Text: DOI References: [1] Camina, R., The Nottingham group, (), 205-221 · Zbl 0977.20020 [2] Gorenstein, D., Finite groups, (1968), Harper and Row New York, London · Zbl 0185.05701 [3] Grigorchuk, R.I., On the Burnside problem for periodic groups, Funct. anal. appl., 14, 41-43, (1980) · Zbl 0595.20029 [4] Grigorchuk, R.I., Just infinite branch groups, (), 121-179 · Zbl 0982.20024 [5] Gupta, N.; Sidki, S., On the Burnside problem for periodic groups, Math. Z., 182, 385-388, (1983) · Zbl 0513.20024 [6] Robinson, D.J.S., A course in group theory, (1982), Springer-Verlag New York, Heidelberg, Berlin · Zbl 0496.20038 [7] Wilson, J.S., Groups with every proper quotient finite, Proc. Cambridge philos. soc., 69, 373-391, (1971) · Zbl 0216.08803 [8] Wilson, J.S., Profinite groups, (1998), Clarendon Press Oxford · Zbl 0909.20001 [9] Wilson, J.S., On abstract and profinite just infinite groups, (), 181-203 · Zbl 0981.20021 [10] Wilson, J.S., On exponential growth and uniformly exponential growth for groups, Invent. math., 155, 287-303, (2004) · Zbl 1065.20054 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.