The same can be said for any abelian group of order 15, leading to the remarkable conclusion that all abelian groups of order 15 are isomorphic. Additive group and Multiplicative group. Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier. Views Read Edit View history. Abelian groups generalize the arithmetic of addition of integers. This article includes a list of references , but its sources remain unclear because it has insufficient inline citations.

Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier. In the case of finitely generated free abelian groups, the proof is easier, does not need the axiom of choice, and leads to a more precise result. A subgroup is an abelian-quotient subgroup i. Given this, the fundamental theorem shows that to compute the automorphism group of G it suffices to compute the automorphism groups of the Sylow p -subgroups separately that is, all direct sums of cyclic subgroups, each with order a power of p. Privacy policy About Groupprops Disclaimers Mobile view.

The term abelian group comes from Niels Henrick Abel, a mathematician who worked with groups even before the formal theory was laid down, in order to prove unsolvability of the quintic. This article includes a list of references , but its sources remain unclear because it has insufficient inline citations. Another way to represent an element of a free abelian group is as a function from B to the integers with finitely many nonzero values; for this functional representation, the group operation is the pointwise addition of functions. Articles lacking in-text citations from September All articles lacking in-text citations CS1: A basis is a subset such that every element of the group can be found by adding or subtracting basis elements, and such that every element's expression as a linear combination of basis elements is unique.

They are named after early 19th century mathematician Niels Henrik Abel. Finitely generated abelian group. September Learn how and when to remove this template message. The classification theorems for finitely generated, divisible, countable periodic, and rank 1 torsion-free abelian groups explained above were all obtained before and form a foundation of the classification of more general infinite abelian groups. Archived from the original on 1 July From Wikipedia, the free encyclopedia. A typical example is the classification of finitely generated abelian groups which is a specialization of the structure theorem for finitely generated modules over a principal ideal domain.

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