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The set is called the underlying set of the group, and the operation is called the group operation or the group law. A group and its underlying set are thus two different mathematical objects. To avoid cumbersome notation, it is common to abuse notation by using the same symbol to denote both. This reflects also an informal way of thinking ...
In mathematics, the group algebra can mean either A group ring of a group over some ring. A group algebra of a locally compact group This page was last edited on 26 ...
Another non-connected group are orthogonal group in even dimension (the determinant gives a surjective morphism to ). More generally every finite group is an algebraic group (it can be realised as a finite, hence Zariski-closed, subgroup of some by Cayley's theorem). In addition it is both affine and projective.
See Rubik's Cube group. In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout ...
In less formal terms, the group consists of words in the generators and their inverses, subject only to canceling a generator with an adjacent occurrence of its inverse. If G is any group, and S is a generating subset of G, then every element of G is also of the above form; but in general, these products will not uniquely describe an element of G.
Let be a group, written multiplicatively, and let be a ring. The group ring of over , which we will denote by [], or simply , is the set of mappings : of finite support (() is nonzero for only finitely many elements ), where the module scalar product of a scalar in and a mapping is defined as the mapping (), and the module group sum of two mappings and is defined as the mapping () + ().
If the group operation is denoted as a multiplication, the order of an element a of a group, is thus the smallest positive integer m such that a m = e, where e denotes the identity element of the group, and a m denotes the product of m copies of a. If no such m exists, the order of a is infinite.
The center of the multiplicative group of non-zero quaternions is the multiplicative group of non-zero real numbers. Using the class equation, one can prove that the center of any non-trivial finite p-group is non-trivial. If the quotient group G/Z(G) is cyclic, G is abelian (and hence G = Z(G), so G/Z(G) is trivial).