4 Lagrange’s Theorem, Normal Subgroups and Quotients
· Denis Auroux
Cosets (recap)
Last time we discussed equivalence relations.
Definition 4.1 (Cosets). Given a subgroup of a group :
is called the (left) coset of containing .
The relation is an equivalence relation on , whose equivalence classes are the left cosets.
The quotient (the set of left cosets) is denoted by . We have a partition
Proof.
, so for all .
If , then , hence , hence .
If and , then and , so , i.e. .
Also, such that such that .∎
Example 4.2. The cosets of are = congruence classes mod (). So
Lagrange’s theorem
Definition 4.3. The index of the subgroup is the number of cosets, .
When is a finite group, since each coset has (bijection ,), the partition implies
Theorem 4.4 (Lagrange’s theorem).
Corollary 4.5. If is a subgroup of a finite group , then divides .
Corollary 4.6. For all a finite group, the order of divides .
(Recall the order of is the smallest such that , and also the order of the subgroup .)
Corollary 4.7. If is prime, then .
(Indeed, take such that ; then has order , hence ,, and by mapping .)
Quotient groups
In the case of , the equivalence relation is compatible with addition, so inherits a group structure, given by addition of cosets:
(the same coset no matter how we choose within their cosets).
In general, trying to define a group structure on by setting might not work. The obstacle to this is: assume () and (). Does it follow that ()? If not, our operation isn’t well defined.
This works fine whenever is the kernel of a homomorphism. Let be a homomorphism with . Recall
Now, if and , then , so ✓. We can define multiplication of cosets by .
In fact, we have a bijection
and this bijection is a group isomorphism! (Because .)
But it fails for general subgroups.
Example 4.8. = symmetries of the square, where = horizontal flip. Then (the coset ). But setting = rotation by , is the reflection across one diagonal, while the coset of is , where is the reflection across the other diagonal.
So , even though (and ).
Normal subgroups
Right cosets vs. left cosets: similarly to the left cosets (), we define right cosets, which correspond to .
Remark. None of these are subgroups of (except for itself)! (They don’t contain !)
Also denote (this one is a subgroup).
Definition 4.9. is a normal subgroup if , (“left cosets = right cosets”; this means the two equivalence relations above agree), or equivalently, ,( is closed under conjugation by elements of ).
Example 4.10.
Any subgroup of an abelian group is normal ( ✓).
In , the subgroup ( = horizontal reflection) is not normal: .
Theorem 4.11. Given a group and a subgroup , the following are equivalent:
there exists a group homomorphism (some other group) with ;
is a normal subgroup;
has a group structure given by .
Proof.(1) ⇒ (2). Suppose there exists a homomorphism with . Then for all ,
but also
So for all : is normal.
(2) ⇒ (3). Assume is normal, and define an operation on by .
We need to check that this is well-defined, i.e. and . Equivalently: ,. Using that is normal, , so
It clearly satisfies the group axioms: ; similarly the other axioms follow from the definition of the operation and the fact that is a group.
(3) ⇒ (1). Now, , is clearly a homomorphism with kernel .∎
Remark. If is a group homomorphism, then is a normal subgroup, and factors as
where is an isomorphism, as seen above. Hence
(with the isomorphism given by ).
Example:
= permutations of = symmetries of the triangle
It contains:
= identity, which does nothing; order 1.
three transpositions, which swap two elements: ,, reflections of the triangle; order 2.
two 3-cycles, and rotations by . These have order 3.
(Cycle notation: means .)
Subgroups of . They have order 1, 2, 3 or 6 (and those of order 2 or 3 are necessarily cyclic):
, trivial;
and two others ();
, the subgroup of rotations ();
all of .
and are obviously normal subgroups.
is not normal: its conjugate is
(rotate clockwise, then swap , then rotate counterclockwise swap ).
is normal. It’s the kernel of
Notes not reached in this lecture, covered at the start of {{LINK:L05}}
Definition 4.12. Say a group is simple if it has no normal subgroups other than and .
We use normal subgroups to view as built from hopefully simpler groups and . Simple groups are then the basic building blocks.
Exact sequences
Definition 4.13. A sequence of groups and homomorphisms
is an exact sequence if ,.
This means such that . In particular, = the trivial homomorphism ( for all )().
A short exact sequence is the simplest case,
( is often denoted for multiplicative groups, for additive ones). This means:
is an injective homomorphism;
is a surjective homomorphism;
.
Proposition 4.14. Such an exact sequence exists iff contains a normal subgroup isomorphic to and such that the quotient group is isomorphic to .
(The prototype short exact sequence is .)
Example 4.15. For any groups and ,
Example 4.16.
and
Example 4.17. There exists an exact sequence
but not (there is no normal subgroup of order 2!).