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An Abelian Group is a group that is commutative. That is
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For a subgroup
of an Abelian group , the partition of into left cosets of and the partition of right cosets are the same. -
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 . -
(Fraleigh 11.15) The finite indecomposable Abelian groups are exactly the cyclic groups of prime power order.
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(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.
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Every Abelian Group is Normal.
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The center of a group
is the Abelian subgroup of defined as -
Another important normal subgroup is the Commutator Group.