In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that the group operation * is commutative, so that for all a and b in G, a * b = b * a. Abelian groups are named after Norwegian mathematician Niels Henrik Abel. Groups in which the group operation is not commutative are called non-abelian (or non-commutative). Since the group operation in an abelian group is commutative as well as associative, the value of a product of group elements is independent of the order in which the product is calculated. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, although infinite abelian groups are the subject of current research.
LatestNews
Testimonials
"It's really great working with you guys. I've never dealt with a company that has been as responsive and supportive as you all have been."
David Pickman
Computer Resource Teacher Herricks UFSD
The multiplicative notation is the usual notation for groups, while the additive notation is the usual notation for modules. When studying abelian groups apart from other groups, the additive notation is usually used.
Example
Every cyclic group G is abelian, because if x, y are in G, then xy = aman = am + n = an + m = anam = yx. Thus the integers, Z, form an abelian group under addition, as do the integers modulo n, Z/nZ.
Every ring is an abelian group with respect to its addition operation. In a commutative ring the invertible elements, or units, form an abelian multiplicative group. In particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication.
Every subgroup of an abelian group is normal, so each subgroup gives rise to a quotient group. Subgroups, quotients, and direct sums of abelian groups are again abelian.
Matrices, even invertible matrices, do not form an abelian group under multiplication because matrix multiplication is generally not commutative.
The fundamental Group of the Torus is abelian
A space with a trivial fundamental group (i.e., every loop is homotopic to the constant loop), is called simply connected. For instance, any contractible space, like Euclidean space, is simply connected. The sphere is simply connected, but not contractible. By definition, the universal cover X^~ is simply connected, and loops in X lift to paths in X^~. The lifted paths in the universal cover define the deck transformations, which form a group isomorphic to the fundamental group.
The underlying set of the fundamental group of X is the set of based homotopy classes from the circle to X, denoted [S^1,X]. For general spaces X and Y, there is no natural group structure on [X,Y], but when there is, X is called a co-H-space. Besides the circle, every sphere S^n is a co-H-space, defining the homotopy groups. In general, the fundamental group is non-Abelian. However, the higher homotopy groups are Abelian. In some special cases, the fundamental group is Abelian. For example, the animation above shows that a*b==b*a in the torus. The red path goes before the blue path. The animation is a homotopy between the loop that goes around the inside first and the loop that goes around the outside first.
Since the first integral homology H_1(X,Z) of X is also represented by loops, which are the only one-dimensional objects with no boundary, there is a group homomorphism
alpha:pi_1(X)->H_1(X,Z),
which is surjective. In fact, the group kernel of alpha is the commutator subgroup and alpha is called Abelianization.
li
