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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:
 2e+19 Limits, profinite groups 2e+08 Subgroup theorems; subgroup growth 2e+27 Residual properties and generalizations; residually finite groups
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##### 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
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