Some divisibility properties of the subgroup counting function for free products
| Autor: | Morris Newman, Michael Grady |
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| Rok vydání: | 1992 |
| Předmět: | |
| Zdroj: | Mathematics of Computation. 58:347-353 |
| ISSN: | 1088-6842 0025-5718 |
| DOI: | 10.1090/s0025-5718-1992-1106969-8 |
| Popis: | Let G be the free product of finitely many cyclic groups of prime order. Let M n {M_n} denote the number of subgroups of G of index n . Let C p {C_p} denote the cyclic group of order p , and C p k C_p^k the free product of k cyclic groups of order p . We show that M n {M_n} is odd if C 2 4 C_2^4 occurs as a factor in the free product decomposition of G . We also show that if C 3 3 C_3^3 occurs as a factor in the free product decomposition of G and if C 2 {C_2} is either not present or occurs to an even power, then M n ≡ 0 mod 3 {M_n} \equiv 0\;\bmod \, 3 if and only if n ≡ 2 mod 4 n \equiv 2\;\bmod \, 4 . If, on the other hand, C 3 3 C_3^3 occurs as a factor, and C 2 {C_2} also occurs as a factor, but to an odd power, then all the M n {M_n} are ≡ 1 mod 3 \equiv 1\;\bmod \, 3 . Several conjectures are stated. |
| Databáze: | OpenAIRE |
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