# Math 545 Midterm 2

The flashcards below were created by user Jorge732 on FreezingBlue Flashcards.

1. Definition 4.8
the symmetric group of degree n
• The group consisting of the set of Sn of
• all permutations on A={1,2,3,….,n}, under the operation of permutation multiplication
• is called the symmetric group of degree n.
2. Definition 4.11
A permutation of theta-ele- Sn is called a cycle
if it is
• A permutation of theta-ele- Sn is called
• a cycle if it is of the form (a1,a2,a3,…,an). The length of a cycle is the
• number of elements in it. We call a cycle a k-cycle if there are k elements in
• it. Two cycles are disjoint if they have no common elements in them.
3. 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 even
• permutation if it can be written as a product of even number 2-cycles, and it
• is called an odd permutation if it can be written as a product of an odd number
• of 2-cycles.
4. Definition 5.4
left coset of H in G
is called the right coset
• Let G be a group, H a subgroup of G, and
• a-element-G. Then the se aH={ah|h-ele-H} is called a left coset of H in G, and
• the set Ha={ha|h-ele-H} is called the right coset.
5. Definition 5.10
the number of distinct left cosets of H in G is
called
• Let G be a finite group and H a subgroup
• of G. Then the number of distinct left cosets of H in G is called the index of
• of H in G and is denoted [G:H].
6. 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’
7. 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
8. 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 called
• an isomorphism. Two groups G and G’ are called isomorphic, written GcongruentG’,
• if there exists some isomorphism FEE:GàG’
 Author: Jorge732 ID: 264214 Card Set: Math 545 Midterm 2 Updated: 2014-02-27 08:08:26 Tags: math545 Folders: Description: Midterm 2 Show Answers: