Subgroups and Quotient Groups
Basic Definitions
Generating sets
Definition: Suppose
Some special types of subgroups
See DF Section 2.2
Suppose
- The centralizer
of is the set of elements such that for all . - The normalizer
of is the set of elements such that . In other words, for all . is a normal subgroup if .- The center
of is the the of elements of such that for all . - If
is a homomorphism, the kernel of is the set of such that .
Notice that:
- The center
is a normal subgroup of . and is a normal subgroup of .- The kernel of any homomorphism is a normal subgroup. (In fact, the converse is true as well, as we will see later).
Subgroups from group actions
Suppose that
- The kernel of the action is the set of
such that for all . In other words, the kernel of the action is the kernel of the homomorphism from to corresponding to the action. The kernel of the action is therefore a normal subgroup of G. - If
, the set of elements such that is a subgroup of called the stabilizer of .
Normalizers and centralizers via group actions
One way to think of the normalizer of
Let
If we choose
If we restrict the action of
Cosets
See DF Section 3.1
Definition: Let
Basics
if and only if .- Any two left (right) cosets are in bijection with each other, so if
is finite all cosets of have the same number of elements as . - Two left (right) cosets are either equal or disjoint.
- There is a bijection between the set of left cosets and the set of right cosets of
in given by . - Together, the left (right) cosets of
form a partition of into disjoint sets. - The index of
in , written , is the number of left (right) cosets, if that number is finite; otherwise we say has infinite index in . is normal if and only if for every .
One way to obtain most ofthe key properties of cosets is to observe that the relation
Theorem: (Lagrange) If
Corollary: In a finite group, the order of an element divdes the order of the group.
Quotient Group
If
The key ingredient of this definition is that the product is well defined. In other words, if
Universal property
Let
- The kernel of this map is
. - Let
be any group and let be a homomorphism such that is contained in the kernel of . Then there is a unique homomorphism such that .
The map
Corollary: A subgroup
In fact if