-
An Abelian Group is a group that is commutative. That is
-
For a subgroup
of an Abelian group , the partition of into left cosets of and the partition of right cosets are the same. -
(Fraleigh 38.12) Every finitely generated Abelian group is isomorphic to a group of the form
Where
divides for . -
Proof: Let
. Consider the homomorphism from (Fraleigh 38.8) see here. Let . Apply (Fraleigh 38.11) to get a basis for with a basis of the form and is a basis for and divides . Now
by the Fundamental homomorphism theorem. But also
-
-
Fundamental Theorem of Finitely Generated Abelian Groups: Every finitely generated Abelian group
is isomorphic to the direct product of the form Where
are primes, not necessarily distinct, and . The direct product is unique except for rearrangement of the factors. The number of factors of
is the unique Betti Number of . -
Apply (Fraleigh 38.12). Note that for each
there exists a prime power decomposition by the fundamental theorem of arithmetic. Let
be the torsion subgroup of . The Betti number is the rank of which is invariant and thus unique. -
Let
be the subgroup obtained by elements of the torsion subgroup of with prime order . It can be shown that any prime power decomposition of has each subgroup isomorphic to the direct product of those cyclic factors of order some power of . Thus, the decomposition into products of cyclic groups is unique. - This holds because there exists a basis
such that each divides . We also have that by the definition of .
- This holds because there exists a basis
-
Additionally, let
. It can be shown to be a subgroup of . Note that
for any and prime . because Thus if each
. -
To show the uniqueness of the prime power decomposition we have that:
Where
and are sorted in ascending order. Thus . Finally, argue by induction that each
. Consider . For each , we have That is,
gives us two decompositions, one of which has an extra factor over the other. Hence a contradiction -
Finally, to show the decomposition for the whole torsion group
, we note that we can decompose into a direct product of all where is a prime factor of . Each has a unique decomposition therefore so does .
-
-
(Fraleigh 11.15) The finite indecomposable Abelian groups are exactly the cyclic groups of prime power order.
-
(Fraleigh 11.16) If
is a finite Abelian group, and if divides , then has a subgroup of order . -
(Fraleigh 11.17) If
is a not divisible by the square of a prime number, then every Abelian group of order is cyclic. -
The torsion subgroup of Abelian group
is the elements of finite order in . It is denoted An Abelian group is torsion free if
is the only element of finite order (Fraleigh e11.43) Every Abelian group is the direct product of its torsion subgroup and of a torsion-free subgroup.
-
Every Abelian Group is Normal.
-
The center of a group
is the Abelian subgroup of defined as -
Another important normal subgroup is the Commutator Group.
Topics
- p-Group and the Sylow Theorems - certain results about
-groups involve Abelian groups. - Free Abelian Group