# Sylow Subgroups of Locally Finite Groups

## Abstract

The theorems of Sylow are among the most basic in the theory of finite groups, and Hall’s theorems on the existence and conjugacy of Hall π-subgroups occupy a similarly central position in the theory of finite soluble groups. It is therefore natural to ask for what kinds of infinite groups results like them are true, and to what extent other parts of finite group theory can be extended to such groups. The answer to the first question has probably turned out to be more disappointing than was at one time expected, in that it is only under rather severe restrictions that sensible analogues of Sylow’s and Hall’s theorems can be obtained; but this paradoxically rekindles interest in the question in that one may now hope for a reasonably complete classification of the circumstances under which theorems like Sylow’s and Hall’s are true. As regards the second question, there are nevertheless some interesting classes of groups with very civilized Sylow structure, and quite a lot of progress has been made in extending such things as formation theory to these groups. I want mainly to discuss the first question from the classification point of view, but first some background may be of interest.

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## References

- [1]A.O. Asar, “A conjugacy theorem for locally finite groups”,
*J. London Math. Soc*. (2) 6 (1973), 358–360.MathSciNetCrossRefMATHGoogle Scholar - [2]E.C. Dade, “Carter subgroups and Fitting heights of finite solvable groups”,
*Illinois J*. Math. 13 (1969), 449–514.MathSciNetMATHGoogle Scholar - [3]A.D. Gardiner, B. Hartley and M.J. Tomkinson, “Saturated formations and Sylow structure in locally finite groups”,
*J. Algebra*17 (1971), 177–211. MR42#7778.MathSciNetCrossRefMATHGoogle Scholar - [4]C.J. Graddon, “Formation theoretic properties of certain locally finite groups” (PhD thesis, University of Warwick, Coventry, 1971 ).Google Scholar
- [5]P. Hall, Some constructions for locally finite groups“,
*J. London Math. Soc*. 34 (1959), 305–319.MathSciNetCrossRefMATHGoogle Scholar - [6]B. Hartley, “F-abnormal subgroups of certain locally finite groups”,
*Proc. London Math. Soc*. (3) 23 (1971), 128–158.MathSciNetCrossRefMATHGoogle Scholar - [7]B. Hartley, “Sylow subgroups of locally finite groups”,
*Proc. London Math. Soc*. (3) 23 (1971), 159–192.MathSciNetCrossRefMATHGoogle Scholar - [8]B. Hartley, “Sylow theory in locally finite groups”,
*Compositio Math*. 25 (1972), 263–280.MathSciNetMATHGoogle Scholar - [9]B. Hartley, “Sylow p-subgroups and local p-solubility”,
*J. Algebra*23 (1972), 347–369.MathSciNetCrossRefMATHGoogle Scholar - [10]B. Hartley, “Complements, baseless subgroups and Sylow subgroups of infinite wreath products”,
*Compositio Math*. 26 (1973), 3–30.MathSciNetMATHGoogle Scholar - [11]B. Hartley, “A class of modules over a locally finite group I”,
*J. Austral. Math. Soc*. 16 (1973), 431–442.MathSciNetCrossRefMATHGoogle Scholar - [12]B. Hartley and A. Rae, “Finite p-groups acting on p-soluble groups”,
*Bull. London Math. Soc*. 5 (1973), 197–198.MathSciNetCrossRefMATHGoogle Scholar - [13]H. Heineken, “Maximale p-Untergruppen lokal endlicher Gruppen”,
*Arch. Math*. (to appear).Google Scholar - [14]C.H. Houghton, “Ends of groups and baseless subgroups of wreath products”,
*Compositio Math*. (to appear).Google Scholar - [15]Otto H. Kegel and Bertram A.F. Wehrfritz,
*Locally finite groups*(North-Holland Mathematical Library, 3. North-Holland, Amsterdam, 1973 ).Google Scholar - [16]L.G. Kovâcs, B.H. Neumann and H. de Vries, “Some Sylow subgroups”,
*Proc. Roy. Soc. London Ser*. A 260 (1961), 304–316.CrossRefMATHGoogle Scholar - [17]David McDougall, “Soluble groups with the minimal condition for normal subgroups”,
*Math. Z*. 118 (1970), 157–167.MathSciNetCrossRefMATHGoogle Scholar - [18]A. Rae, “Sylow p-subgroups of finite p-soluble groups”,
*J. London Math. Soc*. (2) 7 (1973), 117–123.MathSciNetCrossRefMATHGoogle Scholar - [19]A. Rae, “Local systems and Sylow subgroups in locally finite groups II”,
*Proc. Cambridge Philos. Soc*. (to appear).Google Scholar - [20]B.F. WyHrcoe, “U noeanbHO HoHeHHbIx HOH81HO70 pawa” [Locally finite groups of finite rank],
*Algebra i Logika*10 (1971), 199–225.Google Scholar - [21]M.J. Tomkinson, “Formations of locally soluble FC-groups”,
*Proc. London Math. Soc*. (3) 19 (1969), 675–708. MR41#5501.MathSciNetGoogle Scholar - [22]P.T. Bonesawea [R.T. Vol’vaBev], “p-no,grpynnbi Ccnnoea nonHoA IINHENHON npynnw” [Sylow p-subgroups of the general linear group],
*Izv. Akad. Poule SSSR Ser. Mat*. 27 (1963), 1031–1054.Google Scholar - [23]B.A.F. Wehrfritz, “Sylow theorems for periodic linear groups”,
*Proc. London Math. Soc*. (3) 18 (1968), 125–140.MathSciNetCrossRefMATHGoogle Scholar - [24]B.A.F. Wehrfritz, “Soluble periodic linear groups”,
*Proc. London Math. Soc*. (3) 18 (1968), 141–157.MathSciNetCrossRefMATHGoogle Scholar - [25]B.A.F. Wehrfritz, “Sylow subgroups of locally finite groups with min-p ”, J.
*London Math. Soc*. (2) 1 (1969), 421–427.MathSciNetCrossRefMATHGoogle Scholar - [26]B.A.F. Wehrfritz, “On locally finite groups with min-p ”,
*J. London Math. Soc*. (2) 3 (1971), 121–128.MathSciNetCrossRefMATHGoogle Scholar