Small-index subgroup technique
This is a survey article related to:Sylow's theorem
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Contents
Description
The small-index subgroup technique basically starts off with the assumption that there is a simple non-abelian group satisfying certain conditions (most simply, having a given order ), and then tries to find a contradiction by finding subgroups of smaller index.
There are thus two main steps to the technique:
- Getting hold of a proper subgroup of small index.
- Using this subgroup to derive a contradiction.
Possible contradictions from a proper subgroup of small index
The main result: embedding in a symmetric group
The main result we use is as follows:
This states that if is a proper subgroup of
, and
is a simple non-abelian group, then
is isomorphic to a subgroup of the symmetric group
. The idea is to consider the action on the left coset space of
by left multiplication, and observe that this action gives a nontrivial homomorphism from
to
. Simplicity of
then yields that the homomorphism is injective.
The simplest corollary: order considerations
The most basic corollary of this result involves orders:
Order of simple non-Abelian group divides factorial of index of proper subgroup
This states that if is a proper subgroup of a simple non-abelian group
, and
denotes the index of
in
, then the order of
divides
. This follows directly by combining the previous result with Lagrange's theorem.
This result indicates that a simple non-abelian group cannot have proper subgroups of very small index. For instance, a simple non-abelian group of order cannot have a proper subgroup of index
.
More advanced corollaries
With a little more effort, we can establish the following:
In other words, if is a proper subgroup in
of finite index, and
is simple non-abelian, then
is isomorphic to a subgroup, not just of
, but also of
.
This gives a somewhat stronger order consideration:
Order of simple non-abelian group divides half the factorial of index of proper subgroup
Getting beyond order considerations
Once we have established that our given simple non-abelian group is isomorphic to a subgroup of an alternating group, we need to obtain a contradiction. One method of contradiction involves the use of Sylow numbers. (Note that this is slightly different from the use of Sylow numbers to obtain a subgroup of small index). Here, we may derive a contradiction in one of these, or other, ways:
- There is a prime
for which the largest power of
dividing the order of
is the same as that dividing
, but the
-Sylow number of
is greater than the
-Sylow number of
.
- There is a prime
for which the largest power of
dividing the order of
is the same as that dividing
, but the size of the normalizer of the
-Sylow subgroup in
is greater than the size of the normalizer in
.
Getting hold of a proper subgroup of small index
Using Sylow numbers: the basics
Further information: Sylow number equals index of Sylow normalizer
Let be a finite group of order
, and
be a prime divisor of
. Let
denote the
-Sylow number of
: the number of
-Sylow subgroups of
. Since all the
-Sylow subgroups of
are conjugate, the number
also equals the index of any Sylow normalizer: the normalizer of any
-Sylow subgroup. Thus, small values of
give rise to subgroups of small index.
Specifically, if , we get a normal Sylow subgroup, contradicting the assumption that the group is simple non-Abelian. Thus,
, and we get a proper subgroup of index
. Thus, by facts stated in the previous section, the simple non-Abelian group
is isomorphic to a subgroup of the symmetric group of degree
. Even better,
is isomorphic to a subgroup of the alternating group of degree
.
Order considerations thus put some immediate additional constraints on the Sylow numbers:
- Order of simple non-abelian group divides factorial of every Sylow number
- Order of simple non-abelian group divides half the factorial of every Sylow number
Combining constraints on Sylow numbers
If is a simple non-abelian group of order
, the above facts put additional constraints on the Sylow numbers, eliminating very small values. These constraints combine with the congruence condition on Sylow numbers and the divisibility condition on Sylow numbers. In toto, we have, for any prime divisor
of
with
for
relatively prime to
:
.
Eliminating larger possible Sylow numbers
The small-index subgroup technique eliminates very small possibilities for Sylow numbers. To eliminate the larger possibilities, we need other complementary methods. Some of these include enumeration methods: this works when has prime divisors with very small multiplicities (i.e., the largest power of the prime dividing
is very small).
To eliminate small possibilities that are not very small, we need the more sophisticated techniques that use Sylow numbers in and in the alternating group it is embedded in to derive a contradiction. Some illustrative examples are done below.
Some examples
Elimination using the basic methods
We first discuss some examples of elimination that combine the congruence and divisibility conditions with the order constraints on Sylow numbers:
-
: Here, we have
, but
does not divide
, a contradiction.
-
: Here, we have
, but
does not divide
, a contradiction.
-
: Here, we have
, but
does not divide
, a contradiction.
-
: Here, we have
and
, forcing
. But
does not divide
.