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e. In mathematics, specifically group theory, Cauchy's theorem states that if G is a finite group and p is a prime number dividing the order of G (the number of elements in G), then G contains an element of order p. That is, there is x in G such that p is the smallest positive integer with xp = e, where e is the identity element of G.
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.
In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of a symmetric group. [1] More specifically, G is isomorphic to a subgroup of the symmetric group whose elements are the permutations of the underlying set of G. Explicitly, The homomorphism can also be understood as ...
Computational group theory. In mathematics, computational group theory is the study of groups by means of computers. It is concerned with designing and analysing algorithms and data structures to compute information about groups. The subject has attracted interest because for many interesting groups (including most of the sporadic groups) it is ...
A class of groups is a set theoretical collection of groups satisfying the property that if G is in the collection then every group isomorphic to G is also in the collection. This concept arose from the necessity to work with a bunch of groups satisfying certain special property (for example finiteness or commutativity).
Center (group theory) Set of elements that commute with every element of a group. Cayley table for D 4 showing elements of the center, {e, a 2}, commute with all other elements (this can be seen by noticing that all occurrences of a given center element are arranged symmetrically about the center diagonal or by noticing that the row and column ...
Isoclinism of groups. In mathematics, specifically group theory, isoclinism is an equivalence relation on groups which generalizes isomorphism. Isoclinism was introduced by Hall (1940) to help classify and understand p-groups, although it is applicable to all groups. Isoclinism also has consequences for the Schur multiplier and the associated ...
Group isomorphism. In abstract algebra, a group isomorphism is a function between two groups that sets up a bijection between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. From the standpoint of group theory, isomorphic ...