Group Rank

For any prime number p and any positive integer n, the p^n-rank r_(p^n)(G) of a finitely generated Abelian group G is the number of copies of the cyclic group Z_(p^n) appearing in the Kronecker decomposition of G (Schenkman 1965). The free (or torsion-free) rank r_0(G) of G is the number of copies of Z appearing in the same decomposition. It can be characterized as the maximal number of elements of G which are linearly independent over Z. Since it is also equal to the dimension of Q tensor _ZG as a vector space over Q, it is often called the rational rank of G. Munkres (1984) calls it the Betti number of G.

Most authors refer to r_0(G) simply as the "rank" of G (Kargapolov and Merzljakov 1979), whereas others (Griffith 1970) use the word "rank" to denote the sum r_0(G)+sum_(p,n)r_(p^n)(G). In this latter meaning, the rank of G is the number of direct summands appearing in the Kronecker decomposition of G.

See also


This entry contributed by Margherita Barile

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Griffith, P. A. Infinite Abelian Group Theory. Chicago, IL: University of Chicago Press, p. 21, 1970.Kargapolov, M. I. and Merzljakov, Yu. I. "Rank of an Abelian Group." §7.2 in Fundamentals of the Theory of Groups. New York: pp. 53-54, 1979.Munkres, J. R. Elements of Algebraic Topology. Menlo Park, CA: Addison-Wesley, p. 24, 1984.Schenkman, E. "Rank and Linear Independence." §2.4 in Group Theory. Princeton, NJ: Van Nostrand, pp. 51-56, 1965.


Abelian Group, Betti Number, Burnside Problem, Group Torsion, Quasithin Theorem, Quasi-Unipotent Group

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Group Rank

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Barile, Margherita. "Group Rank." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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