Periodica Mathematica Hungarica

, Volume 9, Issue 3, pp 187–204

# Solving equations in groups: A survey of Frobenius' theorem

• H. Finkelstein
Article

## Abstract

A fundamental result of Frobenius states that in a finite group the number of elements which satisfy the equationxn=1, wheren divides the order of the group, is divisible byn. Here 1 denotes the identity of the group. This theorem and several generalizations were obtained by Frobenius at the turn of the century. These results have stimulated a great amount of interest in counting solutions of equations in groups. This article discusses these results and traces the various developments which these fundamental papers have generated.

LetG be a finite group of order |G|. Leto(g) denote the order ofg(ε G). LetH(s, k)={xεG:k|o(x)| sk} wherea/b meansa dividesb and leth(s,k)=|H(s,k)|. Using this notation the simplest of Frobenius' results states ifn/|G|, then/h(n, 1). The minimum value ofh(n, 1) is discussed in the first section. Various conditions are known to insure thath(n, 1)=n. A long standing conjecture of Frobenius states ifn=h(n, 1) thenH(n, 1) is a subgroup (where of coursen/|G|). This conjecture is valid for solvable groups, as well as for various arithmetic conditions.

In the second section other divisibility conditions arising from Frobenius' Theorem are discussed. One direction covers more general arithmetic divisibility condition. Another direction has a much wider scope, involving a finite number of equations of an unspecified form and is mainly due to P. Hall. Recently some divisibility conditions involving all groups of a given order have been obtained. Divisibility conditions also hold in infinite groups, and for automorphism analogues of element order. In the next section generalizations to group characters relating back to Frobenius are given. Some of these expressions are used in analyzing properties of group representations and have applications in quantum theory. In the last section clear evidence is established for the combinatorial rather than group-theoretic nature of these results. In particular, some recent work of Snapper links the counting of solutions of equations with the cycle indices in combinatorial theory. Counting solutions of equations in the symmetric groups is also discussed.

## AMS (MOS) subject classifications (1970)

Primary 20-02 Secondary 05-02 20D99 05A10

## Key words and phrases

Divisibility conditions Frobenius' Theorem automorphism order characters cycle indices

## References

1. [1]
M. Benard andH. Finkelstein, Some counting theorems for finite groups,Arch. Math. (Basel) 26 (1975), 236–239.MR 51#13034Google Scholar
2. [2]
J. L. Berggren, Finite groups in which every element is conjugate to its inverse,Pacific J. Math. 28 (1969), 289–293, [54–823],MR 39#1539Google Scholar
3. [3]
J. L. Berggren, Solvable and supersolvable groups in which every element is conjugate to its inverse,Pacific J. Math. 37 (1971), 21–28.MR 46#5432Google Scholar
4. [4]
R. Brauer, A characterization of the characters of groups of finite order,Ann. of Math. (2)57 (1953), 357–377.MR 14-844Google Scholar
5. [5]
R. Brauer, On a theorem of Frobenius,Amer. Math. Monthly 76 (1969), 12–15. [211–2917],MR 38#3335Google Scholar
6. [6]
R. M. Bryant andL. G. Kovács, A note on generalized characters,Bull. Austr. Math. Soc. 5 (1971), 265–269.Google Scholar
7. [7]
W. Burnside,The theory of groups of finite order (2nd ed.), Dover, New York, 1955.MR 16-1086Google Scholar
8. [8]
S. N. Černikov, Perenesenie odnoî teoremy Frobeniusa na beskonečnye gruppy (Transportation of a theorem of Frobenius to infinite groups),Mat. Sb. 3 (45) (1938), 413–416.Zbl 19, 200Google Scholar
9. [9]
S. N. Černikov, K teoreme Frobeniusa (On a theorem of Frobenius),Mat. Sb. 4 (46), (1938), 531–539.Zbl 21, 12Google Scholar
10. [10]
S. Chowla, I. N. Herstein andW. K. Moore, On recursions connected with symmetric groups I,Canad. J. Math. 3 (1951), 328–334. [207–2874],MR 13-10Google Scholar
11. [11]
S. Chowla, I. N. Herstein andW. R. Scott, The solution ofx d=1 in symmetric groups,Norske Vid. Selks. Forh. (Trondheim) 25 (1952), 29–31. [207–2875],MR 14-947Google Scholar
12. [12]
J. H. E. Cohn, A condition for a finite group to be cyclic,Proc. Amer. Math. Soc. 32 (1972), 48.MR 44#6811Google Scholar
13. [13]
K. A. Corrádi, A remark on finite groups.Ann. Univ. Sci. Budapest Eötvös, Sect. Math. 11 (1968), 125–128. [211–2918],MR 39#4265Google Scholar
14. [14]
C. Curtis andI. Reiner,Representation theory of finite groups and associative algebras, Interscience, New York, 1962.MR 26#2519Google Scholar
15. [15]
P. E. Dubuque, La généralisation du théorème de Turkin,Mat. Sb. 1 (43) (1936), 603–606.Zbl 15, 248Google Scholar
16. [16]
P. E. Dubuque, Sur le théorème de Frobenius,Mat. Sb 2 (44) (1937), 1247–1253.Zbl 18, 204Google Scholar
17. [17]
P. E. Dubuque (Djubjuk), Teorema, soderžaščaja v sebe teoremy Frobeniusa, Weisnera i Turkina o čisle èlementov dannogo projadka v gruppe (A theorem containing the theorems of Frobenius, Weisner and Turkin on the number of elements of given order in a group),Dokl. Akad. Nauk SSSR 20 (1938), 517–519.Zbl 20, 208Google Scholar
18. [18]
P. E. Dubuque (Djubjuk), O fundamental'noî teoreme Frobeniusa, (On the fundamental theorem of Frobenius),Dokl. Akad. Nauk SSSR 21 (1938), 158–161.Zbl 21, 298Google Scholar
19. [19]
P. E. Dubuque, Sur le nombre des éléments d'un groupe qui vérifient certaines conditions,Mat. Sb. 4 (46), (1938) 515–520.Zbl 21. 12.Google Scholar
20. [20]
P. E. Dubuque (Djubjuk), Obobščenie teorem Frobeniusa i Weisnera (A generalization of the theorems of Frobenius and Weisner),Mat. Sb. 5 (47) (1939), 189–196. [211–2904],MR 1-161Google Scholar
21. [21]
E. End Ein Satz von Frobenius und Solomon,Arch. Math. (Basel) 22 (1971), 241–245.MR 45#405Google Scholar
22. [22]
W. Feit, On a conjecture of Frobenius,Proc. Amer. Math. Soc. 7 (1956), 177–187. [42–604],MR 17-1051Google Scholar
23. [23]
W. Feit,Characters of finite groups, Benjamin, New York, 1967.MR 36#2715Google Scholar
24. [24]
H. Finkelstein, Some numerical results on groups,Acta Math. Acad. Sci. Hungar. 26 (1975), 91–96.MR 51#13036Google Scholar
25. [25]
H. Finkelstein, Numerical relationships in direct products of groups,Acta Math. Acad. Sci. Hungar. 28 (1976), 41–50.Google Scholar
26. [26]
H. Finkelstein, The automorphism-order in finite groups,Period. Math. Hungar. 7 (1976), 11–26.Google Scholar
27. [27]
H. Finkelstein andK. Mandelberg, Solutions of equations in symmetric groups.J. Combinatorial Theory Ser. A. (To appear)Google Scholar
28. [28]
F. G. Frobenius, Über auflösbare Gruppen,Sitzungberichte der Königl. Preuß. Akad. Wissenschaften (Berlin) (1893), 337–345.Google Scholar
29. [29]
F. G. Frobenius, Über endliche Gruppen,Sitzungsberichte der Königl. Preuß. Akad. der Wissenschaften (Berlin) (1895), 81–112.Google Scholar
30. [30]
F. G. Frobenius, Verallgemeinerung des Sylow'schen Satzes,Sitzungsberichte der Königl. Preuß. Akad. der Wissenschaften (Berlin) (1895), 981–993.Google Scholar
31. [31]
F. G. Frobenius, Über auflösbare Gruppen II,Sitzungsberichte der Königl. Preuß. Akad. der Wissenschaften (Berlin) (1895), 1027–1044.Google Scholar
32. [32]
F. G. Frobenius, Über auflösbare Gruppen III,Sitzungsberichte der Königl. Preuß. Akad. der Wissenschaften (Berlin) (1901), 849–875.Google Scholar
33. [33]
F. G. Frobenius, Über einen Fundamentalsatz der Gruppentheorie,Sitzungsberichte der Königl. Preuß. Akad. der Wissenschaften (Berlin) (1903), 987–991.Google Scholar
34. [34]
F. G. Frobenius, Über einen Fundamentalsatz der Gruppentheorie II,Sitzungsberichte der Königl. Preuß. Akad. der Wissenschaften (Berlin) (1907), 428–437.Google Scholar
35. [35]
C. S. Fu, On Frobenius' theorem,Quart. J. Math. Oxford Ser. 17 (1946), 253–256. [211–2909],MR 8-436.Google Scholar
36. [36]
V. I. Grošev, O čisle èlementov gruppy, stepen' kotoryh prinadležit proizvol'nomu množestvu èlementov (On the number of elements of a group, powers of which belong to an arbitrary set of elements),Dokl. Akad. Nauk SSSR 24 (1939), 14–17. [211–2905],MR 2-2Google Scholar
37. [37]
O. Grün, Über die genaue Anzahl von, Elementen gegebener Ordnung in einer endlichen Gruppe,Math. Nachr. 1 (1948), 342–344. [211–2910],MR 10-589Google Scholar
38. [38]
O. Grün, Beiträge zur Gruppentheorie II (Über einen Satz von Frobenius),J. Reine Angew. Math. 186 (1945), 165–169. [42–602],MR 10-504Google Scholar
39. [39]
M. Hall, Jr.,The theory of groups, Macmillan,. New York, 1959,MR 21#1996Google Scholar
40. [40]
M. Hall, Jr. andJ. K. Senior,The groups of order 2n (n<-6), Macmillan, New York, 1964.MR 29#5889Google Scholar
41. [41]
P. Hall, A contribution to the theory of groups of prime power order,Proc. London Math. Soc. (2)36 (1933), 29–95.Zbl 7, 291Google Scholar
42. [42]
P. Hall, On a theorem of Frobenius,Proc. London Math. Soc. (2)40 (1936), 468–501.Zbl 15, 202Google Scholar
43. [43]
44. [44]
I. N. Herstein,Topics in Algebra Blaisdell, New York, 1964.MR 30#2028Google Scholar
45. [45]
R. Higgins andD. Ballew, An equation for finite groups,Amer. Math. Monthly 78 (1971), 274–275, 1119Google Scholar
46. [46]
B. Huppert,Endliche Gruppen I., Springer-Verlag, Berlin, 1967.MR 37#302Google Scholar
47. [47]
I. M. Isaacs, Systems of equations and generalized characters in groups,Canad. J. Math. 22 (1970), 1040–1046.MR 42#3193Google Scholar
48. [48]
E. Jacobstal, Sur le nombre d'éléments du groupe symétriqueS n dont l'ordre est un nombre premier,Norske Vid. Selsk. Forh. (Trondheim) 21 (1949), 49–51. [207–2873],MR 11-639Google Scholar
49. [49]
R. Kochendörffer,Group theory, McGraw-Hill, New York, 1970.Google Scholar
50. [50]
B. A. Krutik, O nekotoryh svoîstvah konečnoî gruppy (On some properties of finite groups),Mat. Sb. 10 (52) (1942) 239–247 [211–2907],MR 7-5Google Scholar
51. [51]
A. A. Kulakoff, Über die Anzahl der eigentlichen Untergruppen und der Elemente von gegebener Ordnung inp-Gruppen,Math. Ann. 104 (1931), 778–793.Zbl 1, 386Google Scholar
52. [52]
A. A. Kulakoff, Einige Bemerkungen zur Arbeit „On a theorem of Frobenius” von P. Hall,Mat. Sb. 3 (45) (1938), 403–405.Zbl 19, 155Google Scholar
53. [53]
S. Lubelski, Verallgemeinerung eines Frobeniusschen gruppentheoretischen Satzes,Actas Acad. Ci. Lima 8 (1945), 133–137. [211–2908],MR 8-13Google Scholar
54. [54]
J. H. McKay, Another proof of Cauchy's group theorem,Amer. Math. Monthly 66 (1959), 119. [211–2915],MR 20#5232Google Scholar
55. [55]
G. A. Miller, Addition to a theorem due to Frobenius,Bull. Amer. Math. Soc. 11 (1904), 6–7Google Scholar
56. [56]
G. A. Miller, Some deductions from Frobenius' Theorem,Proc. Nat. Acad. Sci. U.S.A. 28 (1942), 251–254. [211–2906],MR 4-1Google Scholar
57. [57]
G. A. Miller, H. F. Blichfeldt andL. E. Dickson,Theory and application of finite groups, Dover, New York, 1961.MR 23#A925Google Scholar
58. [58]
L. Moser andM. Wyman, On solutions ofx d=1 in symmetric groups,Canad. J. Math. 7 (1955), 159–168.MR 16-904Google Scholar
59. [59]
C. Parkinson, Ambivalence in alternating symmetric groups,Amer. Math. Monthly 80 (1973), 190–192.MR 47#3494Google Scholar
60. [60]
D. S. Passman,Permutation groups, Benjamin, New York, 1968.MR 38#5908Google Scholar
61. [61]
A. N. Prokof'ev, O fundamental'noî teoreme Frobeniusa (On a fundamental theorem of Frobenius),Dokl. Akad. Nauk SSSR 65 (1949), 801–804. [211–2911],MR 10-677Google Scholar
62. [62]
A. N. Prokof'ev, O fundamental'noî teoreme G. Frobeniusa (On a fundamental theorem of G. Frobenius),Učenye Zapiski Pedagogičeskogo Instituta, Kaluga 1 (1950), 61–116Google Scholar
63. [63]
A. N. Prokof'ev, Ob uslovijah pri kotoryh čislo rešeniî uravnenijax n=1 v gruppe javljaetsja naimen'šim (On conditions for which the number of solutions of the equationx n=1 in a group is minimal),Ukrain. Mat. Z. 4 (1952), 427–430. [211–2913],MR 15-196Google Scholar
64. [64]
V. Pták, On Frobenius' theorem.Časopis Pěst. Mat. 78 (1953), 207–212. (In Czech), [211–2914],MR 17-1181Google Scholar
65. [65]
A. Rudvalis andE. Snapper, Numerical polynomials for arbitrary characters,J. Combinatorial Theory 10 (1971), 145–159.MR 46#239Google Scholar
66. [66]
H. Sachs, Einfacher Beweis des Frobeniusschen Fundamentalsatzes der Gruppentheories für den Fall eines quadratfreien Exponenten,Acta Sci. Math. (Szeged) 21 (1960), 309–310. [211–2916],MR 24#A1311Google Scholar
67. [67]
C. Schogt, Some theorems of Lubelski on group theory,Math. Centrum Amsterdam Rapport ZW-1949-021, 13 pp. (1949). (In Dutch), [211–2912],MR 11-415Google Scholar
68. [68]
W. R. Scott,Group theory, Prentice-Hall, Englewood Cliffs 1964,MR 29#4785Google Scholar
69. [69]
S. K. Sehgal, On P. Hall's generalization of a theorem of Frobenius.,Proc. Glasgow Math. Assoc. 5 (1962), 97–100.MR 25#3079Google Scholar
70. [70]
W. T. Sharp, L. C. Biedenharn, E. de Vries andA. J. van Zanten, On quasiambivalent groups,Canad. J. Math. 27 (1975), 246–255.MR 52#3438Google Scholar
71. [71]
E. Snapper The polynomial of a permutation representation,J. Combinatorial Theory 5 (1968), 105–114. [206–2871],MR 37#3937Google Scholar
72. [72]
E. Snapper, Normal subsets of finite groups,Illinois J. Math. 13 (1969), 155–164. [206–2872],MR 38#3331Google Scholar
73. [73]
L. Solomon, On Schur's index and the solutions ofG n=1 in a finite group,Math. Z. 78 (1962), 122–125. [16–226],MR 25#2125Google Scholar
74. [74]
L. Solomon, The solution of equations in groups,Arch. Math. (Basel) 20 (1969), 241–247. [211–2919],MR 40#2742Google Scholar
75. [75]
T. Szele Über die endlichen Ordnungszahlen zu denen nur eine Gruppe gehört,Comment. Math. Helv. 20 (1947), 265–267. [166–2313],MR 9-131Google Scholar
76. [76]
W. K. Turkin, Généralisation du théorème de Frobenius,C. R. Acad. Sci. Paris 193 (1931), 1059–1061.Zbl.3, 100Google Scholar
77. [77]
A. I. Uzkow, Über ein Theorem von Frobenius,Mat. Sb. 1 (43) (1936), 337–339Zbl. 14, 346Google Scholar
78. [78]
A. J. van Zanten andE. de Vries, On the number of roots of the equationx n=1 in finite groups and related properties,J. Algebra 25, (1973), 475–486.MR 47#3509Google Scholar
79. [79]
A. J. van Zanten andE. de Vries, On the number of classes of a finite group invariant for certain substitutions,Canad. J. Math. 26 (1974), 1090–1097Google Scholar
80. [80]
L. Weisner, Groups in which the normalizer of every element except identity is abelian,Bull. Amer. Math. Soc. 31 (1925), 413–416.Google Scholar
81. [81]
L. Weisner, On the number of elements of a group, which have a power in a given conjugate set,Bull. Amer. Math. Soc. 31 (1925), 492–496.Google Scholar
82. [82]
L. Weisner, A theorem concerning direct products,Bull. Amer. Math. Soc. 33 (1927), 44–45.Google Scholar
83. [83]
H. Wielandt, Über die Existenz von Normalteilern in endlichen Gruppen.,Math. Nachr. 18 (1958), 274–280. [42–605],MR 21#2009Google Scholar
84. [84]
H. Zassenhaus,The theory of groups (2nd ed.). Chelsea, New York, 1958.MR 19-939Google Scholar