Download SAM IV Free Structures

SAM IV
Free
Structures
Lecture 2
Contents
Groups
Free groups
Finite groups
SAM IV
Free Structures
Actions
Monoid actions
Group actions
Lecture 2
SAM — Seminar in Abstract Mathematics [Version 20130305]
is created by Zurab Janelidze at Mathematics Division, Stellenbosch Univeristy
SAM IV
Free
Structures
Lecture 2
Contents
Groups
Free groups
Finite groups
1 Groups
Free groups
Finite groups
Actions
Monoid actions
Group actions
2 Actions
Monoid actions
Group actions
Free groups
SAM IV
Free
Structures
Lecture 2
Contents
Groups
Free groups
Finite groups
Definition
Let (M, +, 0) be a monoid. An element x ∈ M is said to be invertible if
there exists an element y ∈ M, called the inverse of x, such that
x + y = 0 = y + x. A group is a monoid in which any element is invertible.
An abelian group is a group which, as a monoid, is commutative.
Actions
Monoid actions
Group actions
Free groups
Define and construct free groups. Show that the additive group of integers
is a free group over the empty set.
Free abelian groups
Define and construct free abelian groups. Show that the multiplicative
group of positive (nonzero) rational numbers is a free abelian group over
the set of prime numbers.
Finite groups
SAM IV
Free
Structures
Lecture 2
Contents
Groups
Free groups
Finite groups
Actions
Monoid actions
Group actions
Finite groups are not free
In a group, the order of an element x is defined as the smallest
natural number n such that x n = 1 (using multiplicative notation for
the operations in the group) when such n exists, and is defined to be
infinity when it does not exist. Show that in a free group every
element has infinite order. Deduce from this that no finite group,
i.e. no group with finite underlying set, is free.
Classification of groups of order 4
Show that in a finite group, the order of an element x always divides
the order of the group, i.e. the number of elements in the group.
Define isomorphism of groups and describe isomorphism classes of
finite groups whose order is 4.
Monoid actions
SAM IV
Free
Structures
Lecture 2
Definition
Let M = (M, ·, 1) be a monoid. An M-set is a pair (X , ·) where X is a set and ·
is a map
M ×X
Contents
Groups
Free groups
Finite groups
·
/X
such that 1 · x = x and (m · n) · x = m · (n · x) for all m, n ∈ M and x ∈ X . When
such map is given we say that M acts on X .
Actions
Monoid actions
Group actions
Free M-sets
Define and construct free M-sets.
M-sets as homomorphisms of monoids
Let (M, ·, 1) be a monoid. Establish a bijection between M-set structures on a set
X and monoid homomorphisms (M, ·, 1) → (X X , ◦, 1X ). Show that the monoid
homomorphism (M, ·, 1) → (M M , ◦, 1M ) corresponding to the action of the
monoid (M, ·, 1) on its underlying set M given by monoid multiplication is
injective. Make the conclusion that any monoid is isomorphic to a submonoid of a
monoid of endofunctions.
Group actions
SAM IV
Free
Structures
Lecture 2
Definition
A G -set, where G is a group, is the same as a G -set with G being
regarded as a monoid.
Contents
Groups
Free groups
Finite groups
Free G -sets
Define and construct free G -sets.
Actions
Monoid actions
Group actions
Embedding theorem
Show that any group G is isomorphic to a subgroup of the group of
bijections G → G .
Rotation as group action
In a G -set, the orbit of an element x is defined as the set {g · x | g ∈ G }.
Show that orbits form a partition of X . Present rotation of a cartesian
plane about the origin as a group action. Describes orbits of this group
action.