Abstract
The varieties in the title are shown to be precisely the product varieties G p *Ab(d) for some prime p and some positive integer d dividing p−1. Here G p 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.
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Almeida, J., Margolis, S.W., Volkov, M.V.: The pseudovariety of semigroups of triangular matrices over a finite field. Theor. Inform. Appl. 39, 31–48 (2005)
Auinger, K., Steinberg, B.: The geometry of profinite graphs with applications to free groups and finite monoids. Trans. Amer. Math. Soc. 356, 805–851 (2004)
Auinger, K., Steinberg, B.: On power groups and embedding theorems for relatively free profinite monoids. Math. Proc. Camb. Phil. Soc. 138, 211–232 (2005)
Doerk, K., Hawkes, T.: Finite Soluble Groups, De Gruyter Expositions in Mathematics 4, Berlin New York, 1992
Eilenberg, S.: Automata, Languages and Machines, Academic Press, New York, Vol B, 1976
Eilenberg, S., Schützenberger, M.P.: On pseudovarieties. Adv. Math. 19, 413–418 (1976)
Gildenhuys, D., Ribes, L.: Profinite groups and Boolean graphs. J. Pure Appl. Algebra 12, 21–47 (1978)
Gorenstein, D.: Finite Groups, Harper & Row, New York, 1968
Hall, M.: Coset representation in free groups. Trans. Amer. Math. Soc. 67, 421–432 (1949)
Hall, M.: A topology for free groups and related groups. Ann. Math. 52, 127–139 (1950)
Hall, M.: The Theory of Groups, The Macmillan Co., New York, N.Y. 1959
Henckell, K., Margolis, S.W., Pin, J.-E., Rhodes, J.: Ash's type II theorem, profinite topology and Mal'cev products, Part I. Internat. J. Algebra Comput. 1, 411–436 (1991)
Kapovich, I., Myasnikov, A.: Stallings foldings and subgroups of free groups. J. Algebra 248, 608–668 (2002)
Lubotzky, A., van den Dries, L.: Subgroups of free profinite groups and large subfields of . Israel J. Math. 39, 25–45 (1981)
Lubotzky, A.: Combinatorial group theory for pro-p groups. J. Pure Appl. Algebra 25, 311–325 (1982)
Lubotzky, A.: Pro-finite presentations. J. Algebra 242, 672–690 (2001)
Lyndon, R.C., Schupp, P.E.: Combinatorial Group Theory, Springer-Verlag, 1977
Margolis, S.W., Sapir, M., Weil, P.: Closed subgroups in pro-V topologies and the extension problem for inverse automata. Internat. J. Algebra Comput. 11, 405–445 (2001)
Neumann, H.: Varieties of Groups. Springer, Berlin Heidelberg New York, 1967
Nikolov, N., Segal, D.: Finite index subgroups in profinite groups. C. R. Math. Acad. Sci. 337, 303–308 (2003)
Pin, J.-E.: BG = PG, a success story. In: Fountain, J.B. (ed.) Semigroups, Formal Languages and Groups, Kluwer, Dordrecht, 1995, pp. 33–47
Pin, J.-E.: Syntactic semigroups. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of language theory. Vol. I, Springer Verlag, Berlin Heidelberg New York, Chap. 10, 1997, pp. 679–746
Ribes, L., Zalesskii, P.A.: On the profinite topology on a free group. Bull. London Math. Soc. 25, 37–43 (1993)
Ribes, L., Zalesskii, P.A.: The pro-p topology of a free group and algorithmic problems in semigroups. Internat. J. Algebra Comput. 4, 359–374 (1994)
Ribes, L., Zalesskii, P.A.: Pro-p Trees and Applications. In: Du Sautoy, M., Segal, D., Shalev, A. (eds.) New Horizons in pro-p Groups. Birkhäuser, Boston, 2000, pp. 75–119
Ribes, L., Zalesskii, P.A.: Profinite Groups, Springer, Berlin, 2000
Serre, J.-P.: Trees, Springer-Verlag, Berlin Heidelberg New York, 1980
Stallings, J.: Topology of finite graphs. Invent. Math. 71, 551–565 (1983)
Steinberg, B.: Inevitable graphs and profinite topologies: Some solutions to algorithmic problems in monoid and automata theory stemming from group theory. Internat. J. Algebra Comput. 11, 25–71 (2001)
Steinberg, B.: Finite state automata: A geometric approach. Trans. Amer. Math. Soc. 353, 3409–3464 (2001)
Steinberg, B.: Inverse automata and profinite topologies on a free group. J. Pure Appl. Algebra 167, 341–359 (2002)
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The authors gratefully acknowledge the support of NSERC
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Auinger, K., Steinberg, B. Varieties of finite supersolvable groups with the M. Hall property. Math. Ann. 335, 853–877 (2006). https://doi.org/10.1007/s00208-006-0767-2
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DOI: https://doi.org/10.1007/s00208-006-0767-2