# Math 545 Midterm 2

 Definition 4.8 the symmetric group of degree n The group consisting of the set of Sn ofall permutations on A={1,2,3,….,n}, under the operation of permutation multiplicationis called the symmetric group of degree n. Definition 4.11 A permutation of theta-ele- Sn is called a cycle if it is A permutation of theta-ele- Sn is calleda cycle if it is of the form (a1,a2,a3,…,an). The length of a cycle is thenumber of elements in it. We call a cycle a k-cycle if there are k elements init. Two cycles are disjoint if they have no common elements in them. Definition 4.22 A Permutation FEE is called an even permutation if and it is called an odd permutation if A Permutation FEE is called an evenpermutation if it can be written as a product of even number 2-cycles, and itis called an odd permutation if it can be written as a product of an odd numberof 2-cycles. Definition 5.4 left coset of H in G is called the right coset Let G be a group, H a subgroup of G, anda-element-G. Then the se aH={ah|h-ele-H} is called a left coset of H in G, andthe set Ha={ha|h-ele-H} is called the right coset. Definition 5.10  the number of distinct left cosets of H in G is called Let G be a finite group and H a subgroupof G. Then the number of distinct left cosets of H in G is called the index ofof H in G and is denoted [G:H]. Definition 6.2 A map FEE: GàG’ from a group G to a group G’ is called A map FEE: GàG’ from a group G to a group G’is called a homomorphism if                         in G  <--   FEE(ab)=FEE(a)FEE(b)  -->       in G’ Definition 6.10 Let FEE:GàG’ be a homomorphism and let e’ be the identity in G’. Then the kernel of FEE is Let FEE:GàG’ be a homomorphism and let e’be the identity in G’. Then the kernel of FEE is the set {x-ele-G|FEE(x)=e’},and denoted ker FEE Definition 6.16 A homomorphism FEE:GàG’ that is 1-1 and onto is called A homomorphism FEE:GàG’ that is 1-1 and onto is calledan isomorphism. Two groups G and G’ are called isomorphic, written GcongruentG’,if there exists some isomorphism FEE:GàG’ AuthorJorge732 ID264214 Card SetMath 545 Midterm 2 DescriptionMidterm 2 Updated2014-02-27T08:08:26Z Show Answers