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From: John on 24 Jul 2010 15:49
Does the Group Z12 x Z15 x Z10 has subgroup of order 18 ?
How To properly approach this Problem ?

Thanks.
From: Tonico on 24 Jul 2010 16:52
On Jul 24, 10:49 pm, John <to1m...(a)yahoo.com> wrote:
> Does the Group Z12 x Z15 x Z10 has subgroup of order 18 ?
> How To properly approach this Problem ?
>
> Thanks.


Take the only (and cyclic, of course) subgroup C_3 of order 3 of Z_12,
the only sbgp. F_3 of order 3 of Z_15 and the only sbgp. C_2 of order
2 of Z_10, then C_3 x F_3 x C_2 is a sbgp. of order 18
From: John on 24 Jul 2010 17:18
On Jul 24, 11:52 pm, Tonico <Tonic...(a)yahoo.com> wrote:
> On Jul 24, 10:49 pm, John <to1m...(a)yahoo.com> wrote:
>
> > Does the Group Z12 x Z15 x Z10 has subgroup of order 18 ?
> > How To properly approach this Problem ?
>
> > Thanks.
>
> Take the only (and cyclic, of course) subgroup C_3 of order 3 of Z_12,
> the only sbgp. F_3 of order 3 of Z_15 and the only sbgp. C_2 of order
> 2 of Z_10, then C_3 x F_3 x C_2 is a sbgp. of order 18

Thank you.
From: Arturo Magidin on 24 Jul 2010 18:22
On Jul 24, 2:49 pm, John <to1m...(a)yahoo.com> wrote:
> Does the Group Z12 x Z15 x Z10 has subgroup of order 18 ?

Yes. In fact, if G is a finite abelian group, then the converse of
Lagrange's theorem holds: for every d that divides the order of G, G
has at least one subgroup of order d. Since here you have an abelian
group of order 1800, and 18 certainly divides 1800, this group has a
subgroup of order 18.


> How To properly approach this Problem ?
>
> Thanks.

Here's a way to do it constructively; if you do not yet know the
structure theorem for finite (or finitely generated) abelian groups,
then restrict yourself to groups that are finite direct products of
cyclic groups.

One way to prove it in general (which is pretty constructive), is to
first consider the case where you have a cyclic group of prime power
order. Then do the case of an abelian group that is a direct product
of cyclic groups of prime power order.

Then show that any cyclic group can be written as a product of groups
of prime power order. Then go to town.

--
Arturo Magidin
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