[seqfan] Re: Comment on A068982 and The Global Cohen-Lenstra Heuristic

Maximilian Hasler maximilian.hasler at gmail.com
Sun Jan 3 14:08:10 CET 2010

On Sun, Jan 3, 2010 at 5:43 AM, Georgi Guninski <guninski at guninski.com> wrote:
> writing about groups, looks like every group that has a map to Z gives a sequence:
> start with a generator, apply the group operation n-1 times, map to a(n).

This does not mean much, because there are 2 cases :

* the "generator" g (I think you mean just any element)
 is of finite order r, then g^n = g^(n mod r)
 and a(n) will be an(y) periodic sequence of period r

* the generator g is of infinite order,
 then the elements { g^n } are all different,
 and you can get any sequence a(n),
 using the map  f = a o h,
 where h (restricted to {g^n})
 is the injection g^n \mapsto n.

> are there other sequences in OEIS resulting from group operations?

All sequences in OEIS (which have offet <= 1)
arise in that form from the group operation "+"
starting with generator  g=1:
g^n = n (e.g. g^3 = 1+1+1)
a(n) = a( g^n )

(The map from the group to Z is nothing else than the original
map/sequence a().)


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