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proof of Sylow theorems
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(Proof)
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We let be a group of order where and prove Sylow's theorems.
First, a fact which will be used several times in the proof:
Proof. By induction on  . If  then there is no  which divides its order, so the condition is trivial.
Suppose , , and the proposition holds for all groups of smaller order. Then we can consider whether divides the order of the center, .
If it does, then by Cauchy's theorem, there is an element of of order , and therefore a cyclic subgroup generated by ,
, also of order . Since this is a subgroup of the center, it is normal, so
is well-defined and of order . By the inductive hypothesis, this group has a subgroup
of order . Then there is a corresponding subgroup of which has
.
On the other hand, if
then consider the conjugacy classes not in the center. By the proposition above, since is not divisible by , at least one conjugacy class can't be. If is a representative of this class then we have
, and since
,
. But
, since
, so has a subgroup of order , and this is also a subgroup of . 
Proposition 3 The intersection of a Sylow p-subgroup with the normalizer of a Sylow p-subgroup is the intersection of the subgroups. That is,
.
Proof. If  and  are Sylow p-subgroups, consider
 . Obviously
 . In addition, since
 , the second isomorphism theorem tells us that  is a group, and
 .  is a subgroup of  , so
 . But  is a subgroup of  and  is a Sylow p-subgroup, so
 is a multiple of  . Then it must be that  , and therefore  , and so
 . Obviously
 , so
 . 
The following construction will be used in the remainder of the proof:
Given any Sylow p-subgroup , consider the set of its conjugates . Then
for some . Observe that every is a Sylow p-subgroup (and we will show that the converse holds as well). We let act on by conjugation:
This is clearly a group action, so we can consider the orbits of under it; this remains true if we only consider elements from some subset of . Of course, if all is used then there is only one orbit, so we restrict the action to a Sylow p-subgroup . Name the orbits
, and let
be representatives of the corresponding orbits. By the orbit-stabilizer theorem, the size of an orbit is the index of the stabilizer, and under this action the stabilizer of any is just
, so
.
There are two easy results on this construction. If then
. If then
, and since the index of any subgroup of divides ,
.
Proposition 4 The number of conjugates of any Sylow p-subgroup of is congruent to modulo 
In the construction above, let . Then and
for . Since the number of conjugates of is the sum of the number in each orbit, the number of conjugates is of the form
, which is obviously congruent to modulo .
Proposition 5 Any two Sylow p-subgroups are conjugate
Proof. Given a Sylow p-subgroup  and any other Sylow p-subgroup  , consider again the construction given above. If  is not conjugate to  then  for every  , and therefore
 for every orbit. But then the number of conjugates of  is divisible by  , contradicting the previous result. Therefore  must be conjugate to  . 
Proposition 6 The number of subgroups of of order is congruent to modulo and is a factor of 
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"proof of Sylow theorems" is owned by Henry. [ full author list (2) | owner history (2) ]
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(view preamble)
Cross-references: factor, sum, congruent, number, stabilizer, index, orbit-stabilizer theorem, action, subset, orbits, group action, conjugation, act on, converse, remainder, multiple, second isomorphism theorem, addition, normalizer, intersection, divisible, proposition, inductive hypothesis, well-defined, normal, subgroup, generated by, cyclic subgroup, Cauchy's theorem, induction, Sylow p-subgroup, equation, side, right, right hand side, left hand side, class equation, center, conjugacy class, size, divides, proof, Sylow's theorems, order, group
This is version 9 of proof of Sylow theorems, born on 2002-07-22, modified 2006-08-13.
Object id is 3182, canonical name is ProofOfSylowTheorems.
Accessed 7317 times total.
Classification:
| AMS MSC: | 20D20 (Group theory and generalizations :: Abstract finite groups :: Sylow subgroups, Sylow properties, $\pi$-groups, $\pi$-structure) |
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Pending Errata and Addenda
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