Abelian Group

A group for which the elements commute (i.e.,  for all elements 

 and ) is called an Abelian group. 

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 Mathematica, the function AbelianGroup[n1, n2, ...] represents the direct product of the cyclic groups of degrees , , ....

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  of any given group order  is given by writing  as

where the  are distinct prime factors, then

(2)

where  is the partition function, which is implemented 

in Mathematica as FiniteAbelianGroupCount[n].

 The values of  for , 2, ... are 1, 1, 1, 2, 1, 1, 1, 3, 2, ... 

(Sloane's A000688).

The smallest orders for which , 2, 3, ... nonisomorphic Abelian groups exist are 1, 4, 8, 36, 16, 72, 32, 900, 216, 144, 64, 1800, 0, 288, 128, ...

 (Sloane's 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, ... (Sloane's 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, ... 

(Sloane's A046054), which occur for orders 1, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, ... (Sloane's 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 , 

then there exists a single Abelian group of order  (denoted ) 

and no non-Abelian groups.

 If the group order is a prime squared , then there are two Abelian groups (denoted  and . If the group order is a prime cubed , then there are three Abelian groups (denoted , , and ), and five groups total. If the order is a productof two primes  and , then there exists exactly one Abelian group of group order  (denoted ).

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

(3)

where  is the Riemann zeta function.

The numbers of Abelian groups of orders  are given by 1, 2, 3, 5, 6, 7, 8, 11, 13, 14, 15, 17, 18, 19, 20, 25, ... (Sloane's A063966) 

for , 2, .... Srinivasan (1973) has also shown that

(4)

where

			(5)
			(6)

(Sloane's A021002, A084892, and A084893) and  is again the 

Riemann zeta function. Note that Richert (1952) incorrectly gave . 

The sums  can also be written in the explicit forms

			(7)
			(8)
			(9)

DeKoninck and Ivic (1980) showed that

(10)

where

			(11)
			(12)

(Sloane's A084911) is a product over primes  and  is again the partition function.