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In mathematics, an abelian group is a commutative group, i.e. a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel.
There are two main notational conventions for abelian groups -- additive and multiplicative.
| Convention | Operation | Identity | Powers | Inverse | Direct sum/product |
|---|---|---|---|---|---|
| Addition | a + b | 0 | na | −a | G ⊕ H |
| Multiplication | a * b or ab | e or 1 | an | a−1 | G × H |
Every cyclic group G is abelian, because if x, y are in G, then xy = aman = am + n = an + m = anam = yx. In particular, the integers Z (under addition) and the integers modulo n Z/nZ (also under addition).
The real numbers form an abelian group under addition, as do the non-zero real numbers under multiplication. Every commutative ring gives rise to two abelian groups in the same fashion -- the additive group of all elements, and the multiplicative group of invertible elements, or units.
Any subgroup of an abelian group is normalIn mathematics, a normal subgroup ''N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G the element g-1ng is still in N''. The statement N is a normal subgroup of G is written: :. Another way to put t, and hence factor groups can be formed at will. Subgroups, factor groups, productsIn mathematics, given a group G and two subgroups H and K of G one can define the product of H and K denoted by HK as the set of all elements of the form hk for all h in H and k in K''. In general HK is not a subgroup hkh'k' is not of the form hk ; it is and direct sumsIn group theory, a group G is called the direct sum of a set of subgroups H if each H is a normal subgroup of G each distinct pair of subgroups has trivial intersection, and G ; in other words, G is generated by the subgroups H . If G is the direct sum of of abelian groups are again abelian.
To verify that a certain finite groupGroup theory In mathematics, a finite group is a group which has finitely many elements. Some aspects of the theory of finite groups were investigated in great depth in the twentieth century, in particular the local theory, and the theory of solvable grou is indeed abelian, a table (matrix) can be drawn up in the similar fashion to a multiplication tableIn mathematics, a multiplication table is used to define a multiplication operation for an algebraic system. In basic arithmetic A multiplication table (as used to teach schoolchildren multiplication) is a grid where rows and columns are headed by the num, where, if the group is G = {g1 = e, g2, ..., gn} under the operation ⋅, the (i, j)'th entry of this table contains the product gi ⋅ gj. The group is abelian if and only if this table is symmetric about the main diagonal (i.e. if the matrix is a symmetric matrix).
This is true since if the group is abelian, then gi ⋅ gj = gj ⋅ gi. This implies that the (i, j)'th entry of the table equals the (j, i)'th entry - i.e. the table is symmetric about the main diagonal.