Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T13:36:48.415Z Has data issue: true hasContentIssue false

3 - A short primer on profinite groups

Published online by Cambridge University Press:  05 February 2018

By
John S. Wilson
Edited by
Pierre-Emmanuel Caprace  and
Nicolas Monod
Pierre-Emmanuel Caprace
Affiliation:
Université Catholique de Louvain, Belgium
Nicolas Monod
Affiliation:
École Polytechnique Fédérale de Lausanne
Get access

Summary

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
New Directions in Locally Compact Groups , pp. 17 - 36
Publisher: Cambridge University Press
Print publication year: 2018

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] M., Abért. Group laws and free subgroups in topological groups. Bull. London Math. Soc. 37 (2005), 525–534.
[2] K., Auinger and B., Steinberg. A constructive version of the Ribes–Zalesskii product theorem. Math. Z. 250 (2005), 287–297.
[3] B., Baumslag and S. J., Pride. Groups with two more generators than relators. J. London Math. Soc. (2) 17 (1978), 425–426.
[4] A. V., Borovik, L., Pyber and A., Shalev. Maximal subgroups in finite and profinite groups. Trans. Amer. Math. Soc. 348 (1996), 3745–3761.
[5] R., Camina. The Nottingham group. In New horizons in pro-p groups (Birkhäuser, 2000), pp. 205–221.
[6] P. M., Cohn. On the embedding of rings in skew fields. Proc. London Math. Soc. (3) 11 (1961), 511–530.
[7] T., Coulbois. Free product, profinite topology and finitely generated subgroups. Internat. J. Algebra Comput. 11 (2001), 171–184.
[8] E., Detomi and A., Lucchini. Profinite groups with multiplicative probabilistic zeta function. J. London Math. Soc. (2) 70 (2004), 165–181.
[9] J. D., Dixon, M. P. F., du Sautoy, A., Mann and D., Segal. Analytic pro-p groups, 2nd edition (Cambridge University Press, 1999).
[10] M. V., Ershov. The Nottingham group is finitely presented. J. London Math. Soc. (2) 71 (2005), 362–378.
[11] B., Fine and G., Rosenberger. Conjugacy separability of Fuchsian groups and related questions. In Combinatorial group theory, Contemp. Math. 109 (1990), 11–18.
[12] M. D., Fried and M., Jarden. Field arithmetic (Springer-Verlag, 1986).
[13] D., Gildenhuis and L., Ribes. Profinite groups and Boolean graphs. J. Pure Appl. Algebra 12 (1978), 21–47.
[14] R. I., Grigorchuk and J. S., Wilson. A structural property concerning abstract commensurability of subgroups. J. London Math. Soc. (2) 68 (2003), 671–682.
[15] D., Haran and A., Lubotzky. Maximal abelian subgroups of free profinite groups. Math. Proc. Cambridge Philos. Soc. 97 (1985), 51–55.
[16] W. N., Herfort and L., Ribes. Torsion elements and centralizers in free products of profinite groups. J. Reine Angew. Math. 358 (1985), 155–161.
[17] B., Herwig and D., Lascar. Extending partial automorphisms and the profinite topology on free groups. J. Reine Angew. Math. 358 (1985), 155–161.
[18] M., Kassabov and N., Nikolov. Cartesian products as profinite completions. Int. Math. Res. Not. 2006, Art. ID 72947, 1–17.
[19] E. I., Khukhro and P., Shumyatsky. Bounding the exponent of a finite group with automorphisms. J. Algebra 212 (1999), 363–374.
[20] J., Labute. Classification of Demushkin groups. Canad. J. Math. 19 (1967), 106–132.
[21] M., Lazard. Groupes analytiques p-adiques. Inst. Hautes E'tudes Sci. Publ. Math. 26 (1965), 389–603.
[22] C. R., Leedham-Green and S. M., McKay. The structure of groups of prime power order. London Mathematical Society Monographs, New Series, 27 (Oxford University Press, 2002).
[23] C. R., Leedham-Green and M. F., Newman. Space groups and groups of primepower order I. Arch. Math. (Basel) 35 (1980), 193–202.
[24] A., Lubotzky and A., Mann. Powerful p-groups. I. finite groups, II. p-adic analytic groups. J. Algebra 105 (1987), 484–505 and 506–515.
[25] A., Lubotzky and J. S., Wilson. An embedding theorem for profinite groups. Arch. Math. (Basel) 42 (1984), 397–399.
[26] W., Magnus. Über diskontinuierliche Gruppen mit einer definierenden Relation (Der Freiheitssatz). J. Reine Angew. Math. 163 (1930), 141–165.
[27] A., Mann. Positively finitely generated groups. Forum Math. 8 (1996), 429–459.
[28] A., Mann. A probabilistic zeta function for arithmetic groups. Internat. J. Algebra. Comput. 15 (2005), 1053–1059.
[29] A., Mann and A., Shalev. Simple groups, maximal subgroups, and probabilistic aspects of profinite groups. Israel J. Math. 96 (1996), 449–468.
[30] C., Martinez López and E. I., Zel'manov. Products of powers in finite simple groups. Israel J. Math. 96 (1996), 449–468.
[31] N., Nikolov and D., Segal. On finitely generated profinite groups. I. Strong completeness and uniform bounds, II. Products in quasisimple groups. Ann. of Math. (2) 165 (2007), 171–238 and 239–273.
[32] L., Ribes and P. A., Zalesskii. On the profinite topology on a free group. Bull. London Math. Soc. 25 (1993), 37–43.
[33] L., Ribes and P. A., Zalesskii. Conjugacy separability of amalgamated free products of groups. J. Algebra 179 (1996), 751–774.
[34] L., Ribes and P. A., Zalesskii. Pro-p trees and applications. In New horizons in pro-p groups (Birkhaüser, 2000), pp. 75–119.
[35] N. S., Romanovskii. Free subgroups of finitely presented groups. Algebra and Logic 16 (1977), 62-68.
[36] N. S., Romanovskii. A generalized theorem on freedom for pro-p groups. Siberian Math. J. 27 (1986), 267–280.
[37] N. S., Romanovskii. Shmel'kin embeddings for abstract and profinite groups. Algebra and Logic 38 (1999), 326–334.
[38] N. S., Romanovskii and J. S., Wilson. Free product decompositions in certain images of free products of groups. J. Algebra 310 (2007), 57–69.
[39] J., Saxl and J. S., Wilson. A note on powers in simple groups. Math. Proc. Cambridge Philos. Soc. 122 (1997), 91–94.
[40] J.–P., Serre. Galois cohomology (Springer-Verlag, 1965).
[41] M. G., Smith and J. S., Wilson. On subgroups of finite index in compact Hausdorff groups. Arch. Math. (Basel) 80 (2003), 123–129.
[42] J. S., Wilson. Polycyclic groups and topology. Rend. Sem. Math. Fis. Milano 51 (1981), 17–28.
[43] J. S., Wilson. On the structure of compact torsion groups. Monatsh. Math. 96 (1983), 57–66.
[44] J. S., Wilson. Profinite groups (Clarendon Press, Oxford, 1998).
[45] J. S., Wilson. On abstract and profinite just infinite groups. In New horizons in pro-p groups (Birkhaüser, 2000), pp. 181–203.
[46] J. S., Wilson. On growth of groups with few relators. Bull. London Math. Soc. 36 (2004), 1–2.
[47] J.S., Wilson. The probability of generating a soluble subgroup of a finite group. J. London Math. Soc. (2) 75 (2007), 431–446.
[48] J. S., Wilson. Large hereditarily just infinite groups. J. Algebra 324 (2010), 248–255.
[49] J. S., Wilson. Free subgroups in groups with few relators. Enseign. Math. 56 (2010), 173–185.
[50] J. S., Wilson. Finite index subgroups and verbal subgroups in profinite groups. Ast'erisque 339 (2011), 387–408.
[51] J. S., Wilson and P. A., Zalesskii. Conjugacy separability of certain torsion groups. Arch. Math. (Basel) 68 (1997) 441–449.
[52] J. S., Wilson and E. I., Zel'manov. Identities for Lie algebras of pro-p groups. J. Pure Appl. Algebra 81 (1992), 103–109.
[53] P. A., Zalesskii and O. V., Mel'nikov. Subgroups of profinite groups acting on trees. Math. USSR-Sb. 63 (1989), 405–424.
[54] P. A., Zalesskii and O. V., Mel'nikov. Fundamental groups of graphs of groups. Leningrad Math. J. 1 (1989), 921–940.
[55] E. I., Zel'manov. On periodic compact groups. Israel J. Math. 77 (1992), 83–95.

Save book to Kindle

To save this book to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

Available formats
×