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
(1) |
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,
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 numbe
(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
.
(10) |
DeKoninck and Ivic (1980) showed that
where
(11) | |||
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