In mathematics, especially in the study of infinite groups, the Hirsch–Plotkin radical is a subgroup describing the normal locally nilpotent subgroups of the group. It was named by Gruenberg (1961) after Kurt Hirsch and Boris I. Plotkin, who proved that the join of normal locally nilpotent subgroups is locally nilpotent; this fact is the key ingredient in its construction.[1][2][3]
The Hirsch–Plotkin radical is defined as the subgroup generated by the union of the normal locally nilpotent subgroups (that is, those normal subgroups such that every finitely generated subgroup is nilpotent). The Hirsch–Plotkin radical is itself a locally nilpotent normal subgroup, so is the unique largest such.[4] In a finite group, the Hirsch–Plotkin radical coincides with the Fitting subgroup but for infinite groups the two subgroups can differ.[5] The subgroup generated by the union of infinitely many normal nilpotent subgroups need not itself be nilpotent,[6] so the Fitting subgroup must be modified in this case.[7]
References
- ^ Gruenberg, K. W. (1961), "The upper central series in soluble groups", Illinois Journal of Mathematics, 5 (3): 436–466, doi:10.1215/ijm/1255630890, MR 0136657.
- ^ Hirsch, Kurt A. (1955), "Über lokal-nilpotente Gruppen", Mathematische Zeitschrift, 63: 290–294, doi:10.1007/bf01187939, hdl:10338.dmlcz/100791, MR 0072874, S2CID 123542672.
- ^ Plotkin, B. I. (1954), "On some criteria of locally nilpotent groups", Uspekhi Matematicheskikh Nauk, New Series, 9 (3(61)): 181–186, MR 0065559.
- ^ Robinson, Derek (1996), A Course in the Theory of Groups, Graduate Texts in Mathematics, vol. 80, Springer, p. 357, ISBN 9780387944616.
- ^ Gray, Mary W. (1970), A radical approach to algebra, Addison-Wesley series in mathematics, vol. 2568, Addison-Wesley, p. 125,
For finite groups this radical coincides with the Fitting subgroup
. - ^ Scott, W. R. (2012), Group Theory, Dover Books on Mathematics, Courier Dover Publications, p. 166, ISBN 9780486140162.
- ^ Ballester-Bolinches, A.; Pedraza, Tatiana (2003), "Locally finite groups with min-p for all primes p", Groups St. Andrews 2001 in Oxford. Vol. I, London Math. Soc. Lecture Note Ser., vol. 304, Cambridge Univ. Press, Cambridge, pp. 39–43, doi:10.1017/CBO9780511542770.009, MR 2051515. See p. 40: "In general the Fitting subgroup in an infinite group gives little information about the structure of the group".
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