Some divisibility properties of the subgroup counting function for free products
HTML articles powered by AMS MathViewer
- by Michael Grady and Morris Newman PDF
- Math. Comp. 58 (1992), 347-353 Request permission
Abstract:
Let G be the free product of finitely many cyclic groups of prime order. Let ${M_n}$ denote the number of subgroups of G of index n . Let ${C_p}$ denote the cyclic group of order p , and $C_p^k$ the free product of k cyclic groups of order p . We show that ${M_n}$ is odd if $C_2^4$ occurs as a factor in the free product decomposition of G . We also show that if $C_3^3$ occurs as a factor in the free product decomposition of G and if ${C_2}$ is either not present or occurs to an even power, then ${M_n} \equiv 0\;\bmod 3$ if and only if $n \equiv 2\;\bmod 4$ . If, on the other hand, $C_3^3$ occurs as a factor, and ${C_2}$ also occurs as a factor, but to an odd power, then all the ${M_n}$ are $\equiv 1\;\bmod 3$ . Several conjectures are stated.References
- S. Chowla, I. N. Herstein, and W. K. Moore, On recursions connected with symmetric groups. I, Canad. J. Math. 3 (1951), 328–334. MR 41849, DOI 10.4153/cjm-1951-038-3
- I. M. S. Dey, Schreier systems in free products, Proc. Glasgow Math. Assoc. 7 (1965), 61–79 (1965). MR 188279
- C. Godsil, W. Imrich, and R. Razen, On the number of subgroups of given index in the modular group, Monatsh. Math. 87 (1979), no. 4, 273–280. MR 538760, DOI 10.1007/BF01637030
- Michael Grady and Morris Newman, Counting subgroups of given index in Hecke groups, A tribute to Emil Grosswald: number theory and related analysis, Contemp. Math., vol. 143, Amer. Math. Soc., Providence, RI, 1993, pp. 431–436. MR 1210530, DOI 10.1090/conm/143/01009
- Marshall Hall Jr., Subgroups of finite index in free groups, Canad. J. Math. 1 (1949), 187–190. MR 28836, DOI 10.4153/cjm-1949-017-2
- W. W. Stothers, The number of subgroups of given index in the modular group, Proc. Roy. Soc. Edinburgh Sect. A 78 (1977/78), no. 1-2, 105–112. MR 480341, DOI 10.1017/S0308210500009860
- W. W. Stothers, Subgroups of finite index in a free product with amalgamated subgroup, Math. Comp. 36 (1981), no. 154, 653–662. MR 606522, DOI 10.1090/S0025-5718-1981-0606522-9
- Wilfried Imrich, On the number of subgroups of given index in $\textrm {SL}_{2}(Z)$, Arch. Math. (Basel) 31 (1978/79), no. 3, 224–231. MR 521474, DOI 10.1007/BF01226441
- K. Wohlfahrt, Über einen Satz von Dey und die Modulgruppe, Arch. Math. (Basel) 29 (1977), no. 5, 455–457 (German). MR 507036, DOI 10.1007/BF01220437 T. Müller, Kombinatorische Aspekte endlich erzeugter virtuell freier Gruppen, Dissertation, Johann Wolfgang Goethe-Universität, 1989.
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Math. Comp. 58 (1992), 347-353
- MSC: Primary 11B50; Secondary 20E06
- DOI: https://doi.org/10.1090/S0025-5718-1992-1106969-8
- MathSciNet review: 1106969