Math 545 Midterm 2
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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.

Definition 4.11
A permutation of thetaele Sn is called a cycle
if it is
 A permutation of thetaele 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 kcycle if there are k elements in
 it. 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 even
 permutation if it can be written as a product of even number 2cycles, and it
 is called an odd permutation if it can be written as a product of an odd number
 of 2cycles.

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
 aelementG. Then the se aH={ahheleH} is called a left coset of H in G, and
 the set Ha={haheleH} 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 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].

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 {xeleGFEE(x)=e’},
 and denoted ker FEE

Definition 6.16
A homomorphism FEE:GàG’ that is 11 and onto is called
 A homomorphism FEE:GàG’ that is 11 and onto is called
 an isomorphism. Two groups G and G’ are called isomorphic, written GcongruentG’,
 if there exists some isomorphism FEE:GàG’