Equivalence of conjugacy and coset definitions of normality
This article gives a proof/explanation of the equivalence of multiple definitions for the term normal subgroup
View a complete list of pages giving proofs of equivalence of definitions
This article gives a proof/explanation of the equivalence of multiple definitions for the term normalizer
View a complete list of pages giving proofs of equivalence of definitions
Contents
Statement
The following are equivalent for a subgroup of a group
and an element
:
-
. Here,
is the conjugate subgroup of
by
.
-
. Here,
is the Left coset (?) of
for the subgroup
and
is the right coset of
for the subgroup
.
The set of such that the equivalent conditions (1) and (2) hold is termed the Normalizer (?) of
in
.
is a Normal subgroup (?) of
if the above two conditions hold for all
.
The techniques used in this proof
For a survey article describing these techniques in more detail, see manipulating equations in groups.
Proof
(1) implies (2)
Given: a subgroup of
,
, and
.
To prove: .
Proof: Informally, we start with:
and multiply both on the right by . We get:
which simplifies to:
.
A clearer justification of the manipulation done on the left side can be obtained by looking at things elementwise. We have . Thus,
.
(2) implies (1)
Given: a subgroup of
,
, and
.
To prove: .
Prof: We start with:
We multiply both sides on the right by , and obtain:
which simplifies to:
.
A clearer justification of the manipulation done on the right side can be obtained by looking at things elementwise. We have and
.