1530300
Description
Flashcards by Kuunani214, updated more than 1 year ago
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Created by Kuunani214
over 9 years ago
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| Question | Answer |
| 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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