Some divisibility properties of the subgroup counting function for free products
Authors:
Michael Grady and Morris Newman
Journal:
Math. Comp. 58 (1992), 347353
MSC:
Primary 11B50; Secondary 20E06
MathSciNet review:
1106969
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Let G be the free product of finitely many cyclic groups of prime order. Let denote the number of subgroups of G of index n . Let denote the cyclic group of order p , and the free product of k cyclic groups of order p . We show that is odd if occurs as a factor in the free product decomposition of G . We also show that if occurs as a factor in the free product decomposition of G and if is either not present or occurs to an even power, then if and only if . If, on the other hand, occurs as a factor, and also occurs as a factor, but to an odd power, then all the are . Several conjectures are stated.
 [1]
S.
Chowla, I.
N. Herstein, and W.
K. Moore, On recursions connected with symmetric groups. I,
Canadian J. Math. 3 (1951), 328–334. MR 0041849
(13,10c)
 [2]
I.
M. S. Dey, Schreier systems in free products, Proc. Glasgow
Math. Assoc. 7 (1965), 61–79 (1965). MR 0188279
(32 #5718)
 [3]
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 (80k:10019), http://dx.doi.org/10.1007/BF01637030
 [4]
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
(94h:11040), http://dx.doi.org/10.1090/conm/143/01009
 [5]
Marshall
Hall Jr., Subgroups of finite index in free groups, Canadian
J. Math. 1 (1949), 187–190. MR 0028836
(10,506a)
 [6]
W.
W. Stothers, The number of subgroups of given index in the modular
group, Proc. Roy. Soc. Edinburgh Sect. A 78
(1977/78), no. 12, 105–112. MR 0480341
(58 #514)
 [7]
W.
W. Stothers, Subgroups of finite index in a free
product with amalgamated subgroup, Math.
Comp. 36 (1981), no. 154, 653–662. MR 606522
(82e:20036), http://dx.doi.org/10.1090/S00255718198106065229
 [8]
Wilfried
Imrich, On the number of subgroups of given index in
𝑆𝐿₂(𝑍), Arch. Math. (Basel)
31 (1978/79), no. 3, 224–231. MR 521474
(80c:20062), http://dx.doi.org/10.1007/BF01226441
 [9]
K.
Wohlfahrt, Über einen Satz von Dey und die Modulgruppe,
Arch. Math. (Basel) 29 (1977), no. 5, 455–457
(German). MR
0507036 (58 #22326)
 [10]
T. Müller, Kombinatorische Aspekte endlich erzeugter virtuell freier Gruppen, Dissertation, Johann Wolfgang GoetheUniversität, 1989.
 [1]
 S. Chowla, I. N. Herstein, and K. Moore, On recursions connected with symmetric groups. I, Canad. J. Math 3 (1951), 328334. MR 0041849 (13:10c)
 [2]
 I. M. S. Dey, Schreier systems in free products, Proc. Glasgow Math. Assoc. 7 (1965), 6179. MR 0188279 (32:5718)
 [3]
 C. Godsil, W. Imrich, and R. Razen, On the number of subgroups of given index in the modular group, Monatsh. Math. 87 (1979), 273280. MR 538760 (80k:10019)
 [4]
 M. Grady and M. Newman, Counting subgroups of given index in Hecke groups, Contemp. Math., Amer. Math. Soc. (to appear). MR 1210530 (94h:11040)
 [5]
 M. Hall, Subgroups of finite index in free groups, Canad. J. Math. 1 (1949), 187190. MR 0028836 (10:506a)
 [6]
 W. W. Stothers, The numbers of subgroups of given index in the modular group, Proc. Roy. Soc. Edinburgh Sect. A 78 (1977), 105112. MR 0480341 (58:514)
 [7]
 , Subgroups of finite index in a free product with amalgamated subgroup, Math. Comp. 36 (1981), 653662. MR 606522 (82e:20036)
 [8]
 W. Imrich, On the number of subgroups of a given index in , Arch. Math. 31 (1978), 224231. MR 521474 (80c:20062)
 [9]
 K. Wohlfahrt, Über einen Satz von Dey und die Modulgruppe, Arch. Math. 29 (1977), 455457. MR 0507036 (58:22326)
 [10]
 T. Müller, Kombinatorische Aspekte endlich erzeugter virtuell freier Gruppen, Dissertation, Johann Wolfgang GoetheUniversität, 1989.
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
11B50,
20E06
Retrieve articles in all journals
with MSC:
11B50,
20E06
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199211069698
PII:
S 00255718(1992)11069698
Article copyright:
© Copyright 1992
American Mathematical Society
