Sylow number

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:

• The number of $p$-Sylow subgroups
• The index of the normalizer of any $p$-Sylow subgroup

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

• Congruence condition on Sylow numbers in terms of maximal Sylow intersection
• Sylow subgroups are in correspondence with Sylow subgroups of quotient by central subgroup: In particular, this implies that the Sylow numbers of a group are the same as the Sylow numbers of the quotient by its center. Thus, for the study of Sylow numbers, it is sufficient to look at centerless groups.

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.