MAT C01
Groups
From Gallian: Chapters 0,1,2
Note: The enumeration in this document is not the same as in the
book
definition 0.1.
Z = cfw_. . . 3, 2, 1, 0, 1, 2, 3, . . . denotes the integers.
N = cfw_1, 2, 3, 4, . . . denotes the natural numbers.
axiom 1

MAT C01
Permutation Groups
From Gallian: Chapter 5
definition 0.1. A permutation of a set X is a bijective mapping of
X to itself. A bijective mapping is a mapping that is one-to-one
and onto.
We are mainly interested in permutations of finite sets, in pa

MAT C01
Group Homomorphisms and Isomorphisms
From Gallian: Chapters 6 and 10(part of)
definition 0.1. Let G and H be groups. A mapping f : G H is
called a (group) homomorphism if for any , G, f () =
f ()f (). (The operation between and on the left hand si

MAT C01
Finite Groups - Subgroups
From Gallian: Chapter 3
Note: The enumeration in this document is not the same as in the
book
definition 0.1.
The order, |G|, of a group G is its cardinality,
i.e. thenumber of elements of G (finite or infinite). If the o

MAT C01
Cyclic Groups
From Gallian: Chapter 4
definition 0.1. Let G be a group.
1. The set cfw_g1, g2, . . . , gn G is called the set of generators of
G if it is the smaller subset of G such that applications of the
group operation on the generators prod