# [seqfan] Re: Is A037289 multiplicative

israel at math.ubc.ca israel at math.ubc.ca
Tue Jul 10 09:35:08 CEST 2012

```Doesn't the Sylow p-subgroup consist of all elements x such that p^m x = 0,
where p^m is the largest power of p in n? Those elements constitute an
ideal of R, since p^m (xy) = (p^m x) y = 0, and therefore in particular a
subring.

Robert Israel
University of British Columbia

On Jul 9 2012, franktaw at netscape.net wrote:

>Looking at this sequence, there is a conjecture that the sequence is
>multiplicative. I can almost prove that it is.
>
>I need a lemma that, if R is a commutative ring of order n, and prime p
>divides n, then there is a Sylow p-subgroup of the additive group of R
>that is in fact a subring if R. It is then easy to see that R must be
>the direct product of these subrings, and the result follows.
>
>(This is an equivalent assertion, by the way; if there is a commutative
>ring with no such "Sylow p-subring", the sequence is not multiplicative
>for that n.)
>
>Does anybody know of such a result? Or can you find a proof?
>