Kronecker Decomposition Theorem

Every finite Abelian group can be written as a group direct product of cyclic groups of prime power group orders. In fact, the number of nonisomorphic Abelian finite groups a(n) of any given group order n is given by writing n as


where the p_i are distinct prime factors, then


where P(n) is the partition function. This gives 1, 1, 1, 2, 1, 1, 1, 3, 2, ... (OEIS A000688).

More generally, every finitely generated Abelian group is isomorphic to the group direct sum of a finite number of groups, each of which is either cyclic of prime power order or isomorphic to Z. This extension of Kronecker decomposition theorem is often referred to as the Kronecker basis theorem.

See also

Abelian Group, Finite Group, Group Order, Kronecker Basis Theorem, Partition Function P

Portions of this entry contributed by Margherita Barile

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Kargapolov, M. I. and Merzljako, Ju. I. Fundamentals of the Theory of Groups. New York: Springer-Verlag, p. 55, 1979.Schenkman, E. Group Theory. Princeton, NJ: Van Nostrand, p. 48, 1965.Sloane, N. J. A. Sequence A000688/M0064 in "The On-Line Encyclopedia of Integer Sequences."

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Kronecker Decomposition Theorem

Cite this as:

Barile, Margherita and Weisstein, Eric W. "Kronecker Decomposition Theorem." From MathWorld--A Wolfram Web Resource.

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