MATH 470 - Exm 2 Verified Solution

MATH 470 - Exam 2 Verified Solutions permutation ️a function from A to A that is both one-to-one and onto permutation group ️a set of permutations of A that forms a group under function composition Every permutation of a finite set can be written as... ️a cycle or as a product of disjoint cycles If the pair of cycles alpha = (a1, a2, ..., am) and beta = (b1, b2, ..., bn) have no entries in common ️Then alpha*beta = beta*alpha The order of a permutation of a finite set written in disjoint cycle form is ️the least common multiple of the lengths of the cycle Sn ️Let A = {1, 2, ..., n} the set of all permutations of A is called the symmetric group of degree n and is denoted by Sn What is the order of Sn ️n! If a permutation a can be expressed as a product of an even (odd) number of 2-cycles ️Then every decomposition of a into a product of 2-cycles must have an even (odd) number of 2-cycles Even permutation ️A permutation that can be expressed as a product of an even number of 2-cycles A set of even permutations in Sn ️Forms a subgroup of Sn An ️The group of even permutations of n symbols, the alternating group of degree n The order of An ️n!/2 isomorphism ️One-to-one onto mapping from G to G' that preserves the group operation. That is f(ab) = f(a)f(b) for all a, b in G Properties of isomorphisms ️1) carries the identity of G to the identity of G' 2) for every integer n and for every group element a, f(a^n) = f(a)^n 3) for any elements a and b in G, a and b commute if and only if f(a) and f(b) commute 4) G = a if and only if G' = f(a) 5) order of a = order of f(a) 6) for a fixed integer k and a fixed group element b in G, the equation x^k = b has the same number of solutions in G as x^k = f(b) in G' 7) if G is finite, then G and G' have exactly the same number of elements of every order. automorphism ️isomorphism from a group G onto itsel inner automorphism ️Let G be a group and let a be an element in G. The function phia defined by phia(x) axa^-1 for all x in G is called the inner automorphism of G induced by a The set of automorphisms of a group and the set of inner automorphisms of a group ️Are both groups under the operation of function composition Aut(Zn) is isomorphic ️To U(n) Aut(G) ️the set of all automorphisms of G Inn(G) ️the set of all inner automorphisms of G Cayley's Theorem ️Every group is isomorphic to a group of permut