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Sunzi's original formulation: x ≡ 2 (mod 3) ≡ 3 (mod 5) ≡ 2 (mod 7) with the solution x = 23 + 105k, with k an integer In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer n by several integers, then one can determine uniquely the remainder of the division of n by the product of these integers, under the condition ...
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 ...
The missionaries and cannibals problem, and the closely related jealous husbands problem, are classic river-crossing logic puzzles. [1] The missionaries and cannibals problem is a well-known toy problem in artificial intelligence, where it was used by Saul Amarel as an example of problem representation. [2][3]
The representation theory of groups divides into subtheories depending on the kind of group being represented. The various theories are quite different in detail, though some basic definitions and concepts are similar. The most important divisions are: Finite groups — Group representations are a very important tool in the study of finite groups.
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 mathematics and 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.
The nominal group technique (NGT) is a group process involving problem identification, solution generation, and decision-making. [1] It can be used in groups of many sizes, who want to make their decision quickly, as by a vote, but want everyone's opinions taken into account (as opposed to traditional voting, where only the largest group is considered). [2]
Restricted representation. In group theory, restriction forms a representation of a subgroup using a known representation of the whole group. Restriction is a fundamental construction in representation theory of groups. Often the restricted representation is simpler to understand. Rules for decomposing the restriction of an irreducible ...