A group action is a binary operation
involving a group and a set, resulting in
another element in that set, where
consecutive group actions are identical to
associating the group elements first, and
then letting it act on the set element.
Consider how one can compute the
a term in Fibonacci
sequence through
,
which can be rewritten as
.
The complexity of computing
in Fibonacci sequence
in this way is if one repeat
such multiplication from up to
. But just as you know, if we
factorize
in binary and compute
,
,
the complexity falls to
drastically. While this is due to the
associativity of matrix multiplication,
it can also be seen as a group action of
acting on ,
and that we are utilizing associativity
of instead of letting
act on
step by step.
Another example of group action we've seen
is permutation group's acting on polynomials,
where we utilize the normality of the
permutation group to construct polynomials
that fit into solution of algebraic
equations, analyzing the solvability of
equation of certain order.
While these examples of group action are
somewhat dedicated to scenarios, in this
text, we are going to see what's the
general view of group action in group
theory, and how we can turn group action
into a powerful tool for studying group
theory, deriving some most fundamental
theorems in group theory: Cauchy's theorem
and the Sylow theorems.
Group actions
For a group and any set , a (left)
group action on is a binary operator
such that
.
Just like all maps will
have properties of group homomorphism
once it fulfils
,
all binary operators will have
properties of group actions once they
fulfil the criteria we showed above.
Orbits and stabilizer subgroups
The orbit
of is the set of the elements
reachable from 's group actions on
. Each element in the orbit is called
a point in the orbit.
's group actions partition into
disjoint orbits: consider equivalence
relation defined by
:
- (Reflexive) .
- (Symmetric) .
- (Transitive) .
And obviously s are the
elements of quotient set .
An element is said to be
fixed by if .
What's more, all elements fixing forms
a subgroup :
- (Identity) .
- (Inverse) .
- (Closed) .
Such a subgroup is called a
stabilizer subgroup of .
For each point ,
it won't be hard to see that
, since
.
Orbit-stabilizer theorem
We know that contains elements
fixing , so outside lies element that
moves . Consider a left coset ,
obviously there's
must be some point distinct from
, and
is one-to-one correspondence between left
cosets of and points in orbit
. It won't be hard to see
.
Such correspondence relationship, alongside
with its derived numberic relationship, is called
the orbit-stabilizer theorem.
When a point is fixed by the entire ,
there's and the orbit
for is trivial. Let's put all elements
with trivial orbit together into group ,
for all non-trivial orbits
,
we get the equation of group action:
The class equation
While can be any set that group can act
on, there's nothing forbidding us to put in
the place of . In fact, doing so turns group
action into one of the most powerful tool for
studying group theory.
Center and centralizer subgroups
Consider a binary operation
, it's
a group action since
and is called the conjugation. Two
elements of are said to be conjugate
when they lies in the same orbit. Each
distinct non-trivial orbit is also called
a conjugacy class.
The center of a group, which is
denoted as ,
is the set of fixed points under conjugation.
Specially, there's :
- .
- .
- .
Noted that
means , is
also the set of elements which commute with
any other element in the group. And since
, there's
.
While is a normal subgroup
rather than some arbitrary set, we can tie
constrains of normal groups to it.
Similarily, for each , we the
stablizer subgroup for under conjugation
as , and call it the
centralizer subgroup. It's obvious that
binary operations of elements in and
elements in conjugacy class corresponding to
commute.
For distinct centralizer subgroups
corresponding to non-trivial conjugacy
classes, which are proper subgroups of ,
we obtain the class equation that:
The class equation imposes numberic constrains
on the size of the center of a group (which is
a normal subgroup) and the number of elements
in a group under conjugation.
Groups of prime-power order have non-trivial center
For a group with ,
is prime, we would like to show its center is
not trivial.
First, for each centralizer subgroup ,
if exists, since it's a proper subgroup,
.
If there's no centralizer subgroup, .
Either case we turn the class equation into a
congruence relation that
.
Then, by ,
is divisible by , it's impossible
for and is not trivial.
Up to this meaning, the case of groups of order
is special and such a group is
sometimes called a -group.
Groups of prime-squared order are abelian
For a group with , is prime,
there's either or .
When , the entire group is
abelian and we are done.
Assume , since has
prime order, there's
.
And for any
,
there's
,
which means any binary operation in commutes
rather than just those with , and is
a contradiction.
So when , we have and
the group is abelian.
Cauchy's theorem
While in the text of cyclic group, we've proved
a special case of
Cauchy's theorem applied to finite abelian groups,
and in this text, we will prove the complete
version of it using the class equation.
The Cauchy's theorem states that given a prime
and a group , if is divisible by ,
then there must be such that .
First, if there exists some centralizer subgroup
whose order is divisible by ,
by , we continue the search
in .
Second, none of the centralizer subgroup has
order divisible by , for each centralizer
subgroup ,
,
and we can turn the class equation into a
congrence relation that
, so does
the case of no centralizer subgroup. Since
divides and is a finite abelian
group, by applying Cauchy's theorem of finite
abelian group we conclude the proof.
A group is called a -group if any
element in the group has order divisible by
prime . And by the Cauchy's theorem, if a
group's order is divisible by prime ,
there must be a subgroup of order which
is cyclic and there's element of order .
So a group is a -group iff its order is
.
The Sylow Theorems
Following the proof of the Cauchy's theorem
and based on the class equation, Sylow made
some development and came up with some
subgroup existence theorems which are very
useful for studying group theory.
First Sylow theorem
The first Sylow theorem moves a step further
from the Cauchy's theorem that if is
divisible by , then there
must be such that .
The proof is done by mathematical induction:
while proving the first Sylow theorem
parameterized by , we assume the theorem
is true when parameterized by
. The case of
is true by the Cauchy's theorem.
First, if there's some centralizer subgroup
with order divisible by , by
, we continue the search
in .
Second, if none of the centralizer subgroup
has order divisible by , then for every
centralizer subgroup , there's
,
and thus ,
so does the case of no centralizer subgroup.
By Cauchy's theorem and 's being prime,
there's some subgroup
. Since 's
binary operations with commute and so do
's, there's
.
Consider the quotient group
, since is
divisible by ,
is divisible by . By applying the
first Sylow theorem parameterized by
and correspondence theorem,
we get some subgroup
of quotient group of order , where
is of order and is the subgroup
we are seeking for.
So for a group such that
, we are able
to find a subgroup such that
, which is maximal in and
called the Sylow -subgroup.
Does it mean we can construct subgroup of
any size using the first Sylow theorem? You see,
starting from a specific size dividing
where are primes, by taking
and ,
since ,
does it mean either or
are the subgroup of size we are
seeking for? The answer is no, since
is not necessarily a group,
there's nothing constraining
to
be still in . Neither does .
They can be a group iff
, however first Sylow theorem has no
guarantee on such property for the
-subgroup and -subgroup retrieved.
Normalizer
In the group , consider binary operation
,
It's a group action on 's subgroups since
and the resulting
is also a subgroup of :
- (Identity) .
- (Inverse) .
- (Closed) .
Such group action is analogous to the
conjugaction of elements, in fact we extend
the concept of conjugation of elements to
subgroups: two subgroups are said to be
conjugate when they lies in the same orbit,
that is, the conjugacy class of subgroups.
Obviously both and are fixed
points by , and subgroups of different
size must lie in differnt conjugacy class.
For each subgroup of , we define its
stabilizer subgroup as and call it
the normalizer of in .
For each subgroup , there's
. First since
,
is a subgroup of . Then since
,
is normal in . And since
contains all elements such that
its left coset equals right coset,
is maximal subgroup in that
is normal in .
For a subgroup , sometimes we
distinguish where we're looking for
normalizer of , so we denote
as the normalizer of in .
And given a subgroup , there's
. Please
notice it is not necessarily the
case , which
it requires to hold, however
we can still have
.
Once you notice is a subgroup
of both deriving properties of
both of them, things become obvious.
Consider the group such that
, and
is prime, let be its Sylow
-subgroup, then for any
of order , if
is fixed by under conjugation,
then .
To prove, when it is trivially
true, and is feasible by the first Sylow
theorem, so we just need to consider
the case of . Since is
fixed by under conjugation, there're
and
.
By the second isomorphism theorem there's
.
Assume ,
there must be , otherwise there're
and .
So there's
,
which is a contradiction to 's being
the Sylow -subgroup, and
is infeasible in this case.
Second Sylow theorem
The second Sylow theorem states that all
Sylow -subgroups are conjugate, that is,
for two Sylow -subgroups of
, ,
or equivalently, under conjugation
.
Let the be a group such that
, and it's
obvious that for
's being Sylow -subgroups.
First, by orbit-stabilizer theorem, there's
while . This is because
so
is divisible by . And if ,
is divisible by then, which
is a contradiction to .
Second, let act on
by conjugation of subgroups and see what will
happen. Let be
the set of fixed points under group action of
, and we have the equation of 's
group action of conjugation (the equation
following orbit-stabilizer theorem, not the
class equation):
Finally, since
,
we can turn the equation into
,
which means
might never be empty. Let's draw some
from the
set, all non-trivial element
has order , and
fixes by conjugation, so ,
and combined with their sharing identity
element we have . Finally
by , we have
, which
concludes our proof.
Third Sylow theorem
By the second Sylow theorem, all Sylow
-subgroups are conjugate and must lie in the
same conjugacy class . For a
group such that ,
the third Sylow theorem states there's
and
dividing .
Following the proof of the second Sylow theorem,
any element taken from
is identical
to , so actually there're
,
and .
According to the orbit-stabilizer theorem,
we have , and
since divides , there can only
be 's dividing .
Following the third Sylow theorem, we would
like to show that a group such that
, where are distinct primes
and , the Sylow -subgroup is
normal in . What's more,
when , there's
,
which means is cyclic.
Consider the Sylow -group in ,
by 's dividing
there can only be
or . And
by ,
, the
only candidate is ,
rendering and
.
Consider the Sylow -group in ,
similarily we have candidates
and ,
and it's possible for the latter case,
e.g. and
. By setting
we eliminate
the latter case so that
and . Since
normalizes each other,
is a subgroup.
And by ,
according to the second isomorphism theorem,
there's
,
finally by the fundamental theorem of finite
abelian groups, is cyclic.
For the example of group of order
but and is not
cyclic, consider the group , there're
but ,
and is not cyclic.
From the proof above, we can clearly see how
group actions and the Sylow theorems augment
our capability of studying much complexer
groups drastically, in a very different way.
Finally, let's linger on the question of taking
subgroup of order following first
Sylow theorem: consider taking a subgroup
of order from , whose
order is . Any group of
order is cyclic, which means if taking
a subgroup of order is feasible, there
must be some element of order
. However after partitioning elements in
into disjoint cycles, the maximal order
of elements in is , formed by a
transposition plus a disjoint -cycle. So
taking a subgroup of order is impossible
in , rendering taking subgroup of
following first Sylow theorem is
infeasible in general.
Conclusion
In this text, we introduce the concept of
group action such that
.
Just like the consequence of group
homomorphism, involving a group exhibits
some symmetric behaviours: this time, a
group action partitions the set on
into disjoint orbits s,
each point in the orbit
corresponds to a left coset
of the stabilizer subgroup of , so
,
which is called the orbit-stabilizer theorem.
By interleaving the points fixed by the
entire into from those by a
proper subgroup of , we derive the
equation of group action that:
Then, we study an group action of a group
acting on itself by conjugation
: points that
are fixed by the whole group forms a
subgroup called the center of the
group, and those fixed by stabilizer
subgroup , which is called the
centralizer subgroup of , forms a
conjugacy class of .
By putting together the center of group
and non-trivial conjugacy classes, we
derive the class equation of that:
The class equation ties the group action
of conjugacy, the center and centralizer
subgroups, and the order of the group
together. With it we can inspect how the
order of group affects the center and
centralizer subgroups, and therefore
constrains the structure of a group.
A classical application of the class
equation is to prove the Cauchy's theorem
of any finite group: for finding a subgroup
of order dividing the order of
the group , either we can find in
of order dividing , which is
a proper subgroup of and the descent
cannot go on indefinitely, or divides
and by is a finite abelian
group we can find in .
Based on the idea of Cauchy's theorem,
Sylow tweaked its process of finding
subgroup and extended its capability to
finding dividing the order of .
First it's possible to find subgroup of
order dividing the order of a group.
And if findinding subgroup of order
of a group is
possible, the subgroup
of order found
in is a cyclic group, and is
normal in . In the quotient group
of , by the
induction we are able to find subgroup
of order ,
and by correspondence theorem, is
the subgroup of order which we
are seeking for. This is also called
the first Sylow theorem.
The second and third Sylow theorems are
about Sylow -subgroups. A Sylow
-subgroup is the maximal subgroup
of order such that does
not divide the order of the group. The
second Sylow theorem states that all Sylow
-subgroups are conjugate, which means
for two Sylow -subgroup
of , there's some such that
, which
can be proved to be 's acting on set
of subgroups of . And the third Sylow
theorem states that given a group such
that , for
the number the Sylow -subgroups
there's
and
dividing .
The proofs of both second and third Sylow
theorem are done by first inspecting the
orbit of a Sylow -subgroups under 's
group action of conjugation on subgroups,
and then choose another Sylow -subgroup
and let it act on the orbit again by
conjugation. The consequence is that the
other Sylow -subgroup lies in the orbit
of the previous one, and is the only one
fixed under the group action of itself in
the second step. So we derive that any two
Sylow -subgroups are conjugate, and
their numberic relations.
So group action is indeed a powerful tool
for utilizing symmetries of a group, and
with appropriate group action chosen (like
conjugation), we even utilize a group's
own symmetries into constraining itself,
which is a quite useful way for studying
group theory.