Abelian Group

An Abelian group is a group for which the elements commute (i.e., AB=BA for all elements A and B). Abelian groups therefore correspond to groups with symmetric multiplication tables.

All cyclic groups are Abelian, but an Abelian group is not necessarily cyclic. All subgroups of an Abelian group are normal. In an Abelian group, each element is in a conjugacy class by itself, and the character table involves powers of a single element known as a group generator.

In the Wolfram Language, the function AbelianGroup[{n1, n2, ...}] represents the direct product of the cyclic groups of degrees n_1, n_2, ....

No general formula is known for giving the number of nonisomorphic finite groups of a given group order. However, 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(k) is the partition function, which is implemented in the Wolfram Language as FiniteAbelianGroupCount[n]. The values of a(n) for n=1, 2, ... are 1, 1, 1, 2, 1, 1, 1, 3, 2, ... (OEIS A000688).

The smallest orders for which n=1, 2, 3, ... nonisomorphic Abelian groups exist are 1, 4, 8, 36, 16, 72, 32, 900, 216, 144, 64, 1800, 0, 288, 128, ... (OEIS A046056), where 0 denotes an impossible number (i.e., not a product of partition numbers) of nonisomorphic Abelian, groups. The "missing" values are 13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 39, 41, 43, 46, ... (OEIS A046064). The incrementally largest numbers of Abelian groups as a function of order are 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, ... (OEIS A046054), which occur for orders 1, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, ... (OEIS A046055).

The Kronecker decomposition theorem states that every finite Abelian group can be written as a group direct product of cyclic groups of prime power group order. If the group order of a finite group is a prime p, then there exists a single Abelian group of order p (denoted Z_p) and no non-Abelian groups. If the group order is a prime squared p^2, then there are two Abelian groups (denoted Z_(p^2) and Z_p×Z_p. If the group order is a prime cubed p^3, then there are three Abelian groups (denoted Z_p×Z_p×Z_p, Z_p×Z_(p^2), and Z_(p^3)), and five groups total. If the order is a product of two primes p and q, then there exists exactly one Abelian group of group order pq (denoted Z_p×Z_q).

Another interesting result is that if a(n) denotes the number of nonisomorphic Abelian groups of group order n, then


where zeta(s) is the Riemann zeta function.

The numbers of Abelian groups of orders <=n are given by 1, 2, 3, 5, 6, 7, 8, 11, 13, 14, 15, 17, 18, 19, 20, 25, ... (OEIS A063966) for n=1, 2, .... Srinivasan (1973) has also shown that



A_k=product_(j=1; j!=k)^(infty)zeta(j/k)
={2.294856591... for k=1; -14.6475663... for k=2; 118.6924619... for k=3,

(OEIS A021002, A084892, and A084893) and zeta(s) is again the Riemann zeta function. Note that Richert (1952) incorrectly gave A_3=114. The sums A_k can also be written in the explicit forms


DeKoninck and Ivic (1980) showed that




(OEIS A084911) is a product over primes p and P(n) is again the partition function.

Bounds for the number of nonisomorphic non-Abelian groups are given by Neumann (1969) and Pyber (1993).

There are a number of mathematical jokes involving Abelian groups (Renteln and Dundes 2005):

Q: What's purple and commutes? A: An Abelian grape.

Q: What is lavender and commutes? A: An Abelian semigrape.

Q: What's purple, commutes, and is worshipped by a limited number of people? A: A finitely-venerated Abelian grape.

Q: What's nutritious and commutes? A: An Abelian soup.

See also

Finite Group, Group Theory, Kronecker Decomposition Theorem, Partition Function P, Ring Explore this topic in the MathWorld classroom

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Arnold, D. M. and Rangaswamy, K. M. (Eds.). Abelian Groups and Modules. New York: Dekker, 1996.DeKoninck, J.-M. and Ivić, A. Topics in Arithmetical Functions: Asymptotic Formulae for Sums of Reciprocals of Arithmetical Functions and Related Fields. Amsterdam, Netherlands: North-Holland, 1980.Erdős, P. and Szekeres, G. "Über die Anzahl abelscher Gruppen gegebener Ordnung und über ein verwandtes zahlentheoretisches Problem." Acta Sci. Math. (Szeged) 7, 95-102, 1935.Finch, S. R. "Abelian Group Enumeration Constants." §5.1 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 273-278, 2003.Fuchs, L. and Göbel, R. (Eds.). Abelian Groups. New York: Dekker, 1993.Kendall, D. G. and Rankin, R. A. "On the Number of Abelian Groups of a Given Order." Quart. J. Oxford 18, 197-208, 1947.Kolesnik, G. "On the Number of Abelian Groups of a Given Order." J. reine angew. Math. 329, 164-175, 1981.Neumann, P. M. "An Enumeration Theorem for Finite Groups." Quart. J. Math. Ser. 2 20, 395-401, 1969.Pyber, L. "Enumerating Finite Groups of Given Order." Ann. Math. 137, 203-220, 1993.Renteln, P. and Dundes, A. "Foolproof: A Sampling of Mathematical Folk Humor." Notices Amer. Math. Soc. 52, 24-34, 2005.Richert, H.-E. "Über die Anzahl abelscher Gruppen gegebener Ordnung I." Math. Zeitschr. 56, 21-32, 1952.Sloane, N. J. A. Sequences A000688/M0064, A063966, and A084911 in "The On-Line Encyclopedia of Integer Sequences."Srinivasan, B. R. "On the Number of Abelian Groups of a Given Order." Acta Arith. 23, 195-205, 1973.

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Abelian Group

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Weisstein, Eric W. "Abelian Group." From MathWorld--A Wolfram Web Resource.

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