Some divisibility properties of the subgroup counting function for free products

Michael Grady, Morris Newman
1992 Mathematics of Computation  
Let G be the free product of finitely many cyclic groups of prime order. Let M" denote the number of subgroups of G of index n . Let Cp denote the cyclic group of order p, and C* the free product of k cyclic groups of order p . We show that Mn is odd if C\ occurs as a factor in the free product decomposition of G. We also show that if C\ occurs as a factor in the free product decomposition of G and if C2 is either not present or occurs to an even power, then Mn = 0 mod 3 if and only if n = 2
more » ... nd only if n = 2 mod 4. If, on the other hand, C| occurs as a factor, and C2 also occurs as a factor, but to an odd power, then all the Mn are s 1 mod 3 . Several conjectures are stated.
doi:10.1090/s0025-5718-1992-1106969-8 fatcat:ovhzj7ziw5dfrnqi4msev2uxvy