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The definition of a group does not require that = for all elements and in . If this additional condition holds, then the operation is said to be commutative, and the group is called an abelian group. It is a common convention that for an abelian group either additive or multiplicative notation may be used, but for a nonabelian group only ...
If a set is such that it cannot be endowed with a group structure, then it is necessarily non-wellorderable. Otherwise the construction in the second section does yield a group structure. However these properties are not equivalent. Namely, it is possible for sets which cannot be well-ordered to have a group structure.
A Lie group is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups are named after Sophus Lie , who laid the foundations of the theory of continuous transformation groups .
Let T be a spanning tree for Y and define the fundamental group Γ to be the group generated by the vertex groups G x and elements y for each edge of Y with the following relations: y = y −1 if y is the edge y with the reverse orientation. y φ y,0 (x) y −1 = φ y,1 (x) for all x in G y. y = 1 if y is an edge in T. This definition is ...
The quotient group is the same idea, although one ends up with a group for a final answer instead of a number because groups have more structure than an arbitrary collection of objects: in the quotient / , the group structure is used to form a natural "regrouping".
In mathematics, a group scheme is a type of object from algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field.
The infinite dihedral group is an infinite group with algebraic structure similar to the finite dihedral groups. It can be viewed as the group of symmetries of the integers. The orthogonal group O(2), i.e., the symmetry group of the circle, also has similar properties to the dihedral groups.
The group of fractions or group completion of a semigroup S is the group G = G(S) generated by the elements of S as generators and all equations xy = z that hold true in S as relations. [11] There is an obvious semigroup homomorphism j : S → G ( S ) that sends each element of S to the corresponding generator.