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Re: What do you call the rank of an abelian group?
Posted:
Jun 26, 2006 12:58 PM
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In article <e7or3q$25vt$1@agate.berkeley.edu>, Arturo Magidin
<magidin@math.berkeley.edu> wrote:
> 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."
Well, of course, it is futile to argue about definitions but lest it be
thought that the one I gave was a product of my fevered imagination, it
is the one given by L. Fuchs (in "Infinite Abelian Groups" Vol. 1 for
example).
For a fixed prime p, the p rank, r_p, is the cardinality of a maximal
linearly indepent set of elements of order a power of p. The torsion
free rank, r_0, the cardinality of a maximal linearly independent set
of elements of infinite order. The rank, r, is
r(G) = r_0(G) + sum(r_p(G), p prime). All of these are invariants of G.
G is indecomposable iff r(G) = 1 (not true for r_0) and is locally
cyclic iff r_0(G) + max(r_p(G)) = 1.
--
Paul Sperry
Columbia, SC (USA)
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