Subgroups
Denition
If (G , ) is a semigroup, then by a subsemigroup of G , we mean
non-empty set H G such that (H , ) itself is a semigroup.
If (G , ) is a group, then by a subgroup of G we mean a
subsemigroup that is also a group.
Note: or < are sometime
Denition
Denition (Permutation of a set )
A permutation of a set A is a function : A A that is both
one-to-one and onto.
Example: Given A = cfw_1, 2, nd all permutations. What about
A = cfw_a, b , c
Composition of permutations
Question) If and are permut
Orbits, Cycles, and the Alternating Groups
In this section, we will study orbits and cycles of permutations, and
dene alternating groups An of Sn .
An orbit of a permutation (of a set A).
First, if is a permutation of a set A, then determines a partition
Groups
In this section, we will dene the concepts of the inverse of an
element, a binary structure called a group, and look at some
examples of groups.
Inverse of an element
Denition
Let (S , ) be a groupoid with an identity element e . Given x S ,
we say
Cosets and the theorem of Lagrange
Denition
Let H be a subgroup of a group G . Then given a G , the subset
aH = cfw_ah | h H of G is the left coset of H containing a, and
Ha = cfw_ha | h H of G is the right coset of H containing a.
Note: If G is abelian
Generating sets
In this section, we will dene a generating set of a group. (We skip
the Cayley diagraphs.)
Generators
Denition
Let G be a group and ai G for i I . If the set S = cfw_ai | i I
generates the (sub)group G, we call the cfw_ai s the the set of
Isomorphic Binary structures
In this section, we will study structure-preserving mappings (called
morphisms) from one structure to another.
Morphisms
Denition
If (G1 , 1 ) and (G2 , 2 ) are two structures (called groupoids), then
by a (homo)morphism from
Cyclic Group
In this section, we will study cyclic groups more in depth.
Recall that if G =< a >, then G = cfw_an | n Z.
Theorem
Every cyclic group is abelian. That is, for all g1 , g2 G cyclic ,
g1 g2 = g2 g1 .
Division algorithm for Z
Theorem
If m Z+ an
Algebraic structure
An algebraic structure, roughly speaking, is a collection of objects
together with a set of rules on how to combine objects.
Each algebraic structure can be regarded as a game with its
own players and its own rules.
Common rules of die