Motivation:
-----------


I'm studying certain properties of conjugation in $SL(n,q)$. There's a nice number, a bit like a covering number, that one can associate with an arbitrary group. In writing a programme in GAP to compute this number, a shortcut comes to mind - I'm not going to share what exactly - that involves the [chief series](https://en.m.wikipedia.org/wiki/Chief_series) of the groups.


The Question:
-------------



> 
> What, if anything, is the maximum length of the chief series of $SL(n,q)$?
> 
> 
> 


Thoughts:
---------


I ran the following in GAP:



```
for n in [2..6] do
   for q in Filtered([2..11],IsPrime) do
         Print([n,q,Size(ChiefSeries(SL(n,q)))],"\n");
   od;
od;

```

It produced . . .



```
[ 2, 2, 3 ]
[ 2, 3, 4 ]
[ 2, 5, 3 ]
[ 2, 7, 3 ]
[ 2, 11, 3 ]
[ 3, 2, 2 ]
[ 3, 3, 2 ]
[ 3, 5, 2 ]
[ 3, 7, 3 ]
[ 3, 11, 2 ]
[ 4, 2, 2 ]
[ 4, 3, 3 ]
[ 4, 5, 4 ]
[ 4, 7, 3 ]
[ 4, 11, 3 ]
[ 5, 2, 2 ]
[ 5, 3, 2 ]
[ 5, 5, 2 ]
[ 5, 7, 2 ]

```

. . . after which I terminated the calculation.


As you can see, the maximum length calculated is $4$ and is obtained by both $SL(2,3)$ and $SL(4,5)$. This is very little to go by, of course.


*What do I think it would be?*


I'd be pleasantly surprised if the maximum existed.




---


Please help :)



Is there a maximum length of the chief series of $SL(n,q)$?
Well, $PSL(n,q)$ is usually simple, so apart from $SL(2,3)$ you're looking at two plus the number of prime factors, counting multiplicity, of the greatest common divisor of $n$ and $q-1$. Of course, this is unbounded.


