Sylow number

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Definition

Definition with symbols

Let G be a finite group and p a prime dividing the order of G. Then the Sylow number for p, denoted as n_p, is any of the following equivalent values:

Equivalence of definitions

Further information: Sylow number equals index of Sylow normalizer

The equivalence of definitions follows from the fact that any two p-Sylow subgroups are conjugate.

Facts

It divides the index of the Sylow subgroup

Further information: Divisibility condition on Sylow numbers

Let m be the index of the Sylow subgroup. Then n_p divides m. The easiest way of seeing this is from the fact that if P is a p-Sylow subgroup, then:

[G:P] = [G:N_G(P)][N_G(P):P]

or:

m = n_p [N_G(P):P]

It is 1 modulo the prime

Further information: Congruence condition on Sylow numbers

This is the congruence condition on the Sylow number. It states that:

n_p \equiv 1 \mod p

This follows from the fact that if we fix any one Sylow subgroup P and look at the orbits on the set Syl_p(G) of all Sylow subgroups under conjugation by elements of P, all the orbits except P itself have sizes as multiples of p.

We can in fact refine the congruence condition further, to obtain certain conditions where we can force n_p to be 1 modulo higher powers of p. The idea in those is to argue that any intersection of Sylow subgroups must have large index in both.

Other facts

Particular cases

Here, we list some finite non-nilpotent groups, along with their Sylow numbers. Note that for finite nilpotent groups (and in particular for finite abelian groups) all the Sylow numbers are equal to 1. In fact, we concentrate on centerless groups because the Sylow numbers of a group are the same as those of its quotient by its center.

Group Order n_2 n_3 n_5 n_7 Other groups admitting it as quotient by center or hypercenter
symmetric group:S3 6 3 1 direct product of S3 and Z2, dicyclic group:Dic12, direct product of S3 and Z3
dihedral group:D10 10 5 1
alternating group:A4 12 1 4 special linear group:SL(2,3), direct product of A4 and Z2
wreath product of Z3 and Z2 18 1 3
generalized dihedral group for E9 18 1 9
symmetric group:S4 24 3 4 general linear group:GL(2,3), binary octahedral group

Using Sylow numbers

Note that the above constraints on Sylow numbers are all constraints that arise purely from the order of the group. Thus, given any positive integer N, we define a set of Sylow numbers for this positive integer as a set (ordered) of associations of n_p to each prime p dividing N such that there exists a group G of order N whose Sylow numbers are precisely n_p.

Of course, given a positive integer, there may be many possibilities for the set of Sylow numbers for that positive integer. Imposing conditions on the kind of group we allow can put further constraints on the set of Sylow numbers. For instance:

  • A set of simple Sylow numbers is a set of Sylow numbers that arises from a simple group.