Abstract Algebra Definitions

Pregunta	Respuesta
Equivalence Relation	An equivalence relation ~ on a set S is on that satisfies these three properties for all x, y, z in S Reflexive: x ~ x Symmetric: x ~ y -> y ~ x Transitive: x ~ y -> y ~ z -> x ~ z
Binary Operation	A binary operation * on a set S is a function mapping S x S into S.
Commutative	A binary operation * on a set S is commutative if and only if a * b = b * a for all a,b in S.
Associative	A binary operation * on a set S is associative if and only if (a * b) * c = a * (b * c) for all a, b, c in S.
Binary Algebraic Structure	A binary algebraic structure (S, ) is a set S together with a binary operation on S.
Identity Element	Let (S, ) be a binary structure. An element e of S is an identity element for if e * s = s * e = s for all s in S. The identity element is unique.
Isomorphism	Let (S, ) and (S', ') be binary algebraic structures. An isomorphism of S with S' is a one-to-one function φ mapping S onto S' such that φ(x * y) = φ(x) *' φ(y) for all x, y in S.
Group	A group (G, ) is a set G, closed under the binary operation , such that the following axioms are satisfied: * is associative There is an identity element e for * in G For each a in G, there is an inverse a' in G such that a * a' = a' * a = e
Abelian	A group G is abelian if its binary operation is commutative.
Left and Right Cancellation Laws	If G is a group with binary operation , then the left and right cancellation laws hold in G, that is, a b = a * c implies b = c and b * a = c * a implies that b = c for all a, b, c in G.
Subgroup	If a subset H of a group G is closed under the binary operation of G and if H with the induced binary operation from G is itself a group, then H is a subgroup of G.
Cyclic Subgroup of G generated by a	Let G be a group and let a be in G. The subgroup {a^n \| n in Z} of G, which is the smallest subgroup of G that contains a, is the cyclic subgroup of G generated by a.
Cyclic Group	A group G is cyclic if there is some element a in G that generates G, that is if <a> = G. a is a generator for G.
Order of a cyclic subgroup	If the cyclic subgroup <a> of G is finite, then the order of a is the order \|<a>\| of this cyclic subgroup. Otherwise it of infinite order.
Permutation of a set	A permutation of a set A is a function φ : A -> A that is both one to one and onto.
Symmetric Group	Let A be the finite set {1, 2, ..., n}. The group of all permutations of A is the symmetric group of n letters, and is denoted S(sub)n
Alternating Group	The subgroup S(sub)n consisting of the even permutations of n letters is the alternating group A(sub)n of n letters.
Left and Right Cosets	Let H be a subgroup of G. The subset aH = {ah \| h is in H} of G is the left coset of H containing a. The subset Ha = {ha \| h is in H} of G is the right coset of H containing a.
Index (G : H) of H in G	Let H be a subgroup of a group G. The number of left cosets of H in G is the index (G:H) of H in G.
Homomorphism	A map φ of a group G into a group G' is a homomorphism if the homomorphism property φ(ab) = φ(a)φ(b) holds for all a, b in G.
Normal	A subgroup H of a group G is normal if its left and right cosets coincide, that is, if gH = Hg for all g in G.
Simple	A group is simple if it is nontrivial and has no proper nontrivial normal subgroups.
Factor (Quotient) Groups	Let H be a normal subgroup of a group G. Then the cosets of H form a factor group (or quotient group) of G by H under the operation (aH)(bH) = (ab)H.

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