Search results
Results From The WOW.Com Content Network
Abelian group. In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian ...
Finitely generated abelian group. Commutative group where every element is the sum of elements from one finite subset. In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements in such that every in can be written in the form for some integers . In this case, we say that the set is a generating set ...
Order p 2: There are just two groups, both abelian. Order p 3: There are three abelian groups, and two non-abelian groups. One of the non-abelian groups is the semidirect product of a normal cyclic subgroup of order p 2 by a cyclic group of order p. The other is the quaternion group for p = 2 and a group of exponent p for p > 2.
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group. [1][2] The commutator subgroup is important because it is the smallest normal subgroup such that the quotient group of the original group by this subgroup is abelian.
The Cayley table tells us whether a group is abelian. Because the group operation of an abelian group is commutative, a group is abelian if and only if its Cayley table's values are symmetric along its diagonal axis. The group {1, −1} above and the cyclic group of order 3 under ordinary multiplication are both examples of abelian groups, and ...
A general exact sequence is sometimes called a long exact sequence, to distinguish from the special case of a short exact sequence. [1] A long exact sequence is equivalent to a family of short exact sequences in the following sense: Given a long sequence. with n ≥ 2, we can split it up into the short sequences.
The structure of any finite abelian group is relatively simple; every finite abelian group is the direct sum of cyclic p-groups. This can be extended to a complete classification of all finitely generated abelian groups, that is all abelian groups that are generated by a finite set. The situation is much more complicated for the non-abelian groups.
Direct sum of groups. In mathematics, a group G is called the direct sum[1][2] of two normal subgroups with trivial intersection if it is generated by the subgroups. In abstract algebra, this method of construction of groups can be generalized to direct sums of vector spaces, modules, and other structures; see the article direct sum of modules ...