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Re: What do you call the rank of an abelian group?
Posted:
Jun 26, 2006 10:31 AM
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In article <260620060152578191%plsperry@sc.rr.com>,
Paul Sperry <plsperry@sc.rr.com> wrote:
>In article
><30307451.1151263755903.JavaMail.jakarta@nitrogen.mathforum.org>, Dror
>Speiser <shin@mycell.org> wrote:
>
>> If an abelian group G has n factors Zi_1*Zi_2*...*Zi_n
>> with i_j dividing i_j+1 then we know G can be represented by n generators,
>> and no less.
>>
>> Do we say G has rank n?
>
>No, consider Z_6 which certainly has one generator but also has a set
>consisting of two independent elements namely 2 and 3 (the fact that 2
>and 3 generate is not important here). The rank of an abelian group is
>the cardinality of a _maximal_ independent set of elements of infinite
>or prime power order. So, Z_6 has rank 2.
>
>[...]
I believe what you describe is called the "Prufer rank" (e.g.,
Robinson's "A Course in the Theory of Groups" 2nd edition, pp. 99).
Rotman ("Introduction to the Theory of Groups", 4th Edition) defines
the rank of a torsion-free group G to be the number of elements r(G)
in a maximal independent subset. Then he defines the rank r(G) of an
arbitrary abelian group to be r(G/G_{tor}), where G_{tor} is the
subgroup of torsion elements; this is the one I am most familiar with.
Rank is also used for absolutely free (and sometimes relatively free
groups) as the number of elements in a basis of F.
For p-groups, I've seen d(G) to represent the number of elements in a
minimal generating set; but I do not recall seeing it called something
other than "the number of elements in a minimal generating set."
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================
Arturo Magidin
magidin@math.berkeley.edu
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