Varieties of finite supersolvable groups with the M. Hall property |
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Authors: | Karl Auinger Benjamin Steinberg |
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Institution: | 1. Fakult?t für Mathematik, Universit?t Wien, Nordbergstrasse 15, 1090, Wien, Austria 2. School of Mathematics and Statistics, Carleton University, Ottawa, Ontario, K1S 5B6, Canada
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Abstract: | The varieties in the title are shown to be precisely the product varieties Gp*Ab(d) for some prime p and some positive integer d dividing p−1. Here Gp denotes the variety of all finite p-groups and Ab(d) the variety of all finite Abelian groups of exponent dividing d. It turns out that these are exactly those varieties H of supersolvable groups for which all finitely generated free pro-H groups are freely indexed in the sense of Lubotzky and van den Dries. Several alternative characterizations of these varieties
are presented. Some applications to formal language theory and finite monoid theory are also given. Among these is the determination
of all supersolvable solutions H to the equations PH = J*H and J*H = J H which is, to the present date, the most complete solution to a problem raised by Pin. Another consequence of our results
is that for each such variety H the monoid variety PH = J*H = J H has decidable membership.
The authors gratefully acknowledge the support of NSERC |
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Keywords: | 20E18 20E10 20D10 20E08 20M07 |
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