Some trivia on derived subgroups
While studying for my exam on Computational group theory I came up with some random questions about derived subgroups. Recall that the derived subgroup of a group is the group generated by the commutators. This group plays an important role in many aspects of group theory:
- if it is again equal to we call this group perfect, which means that it is very far from being abelian, all non-abelian finite simple groups satisfy this property but there are others, the first one being which of course has a normal subgroup isomorphic to and the quotient gives ;
- using the fact that special linear groups are perfect we can prove that is simple for all values of and except and ;
- if you iteratively construct the derived subgroup and end up with a trivial group we say is solvable, which has a link to Galois theory;
- a similar construction yields the notion of nilpotency.
An important part of the definition is that the derived subgroup is necessarily generated by commutators. It is not equal to the set of commutators. There is no reason to assume it should be, but there is none either to assume otherwise. At least I don’t see any reason for this, and given that there’s a MathOverflow question dedicated to this I assume I shouldn’t feel embarrassed. There is a link to a related Math.SE question, which gives us the answer to the question “What is the first example of this behavior?“.
My next question was “How does this continue?” Using a little GAP code to explore all groups of order up to 1023 I got some more examples. An overview of the results:
|96||2||isomorphic to or |
|144||3||isomorphic to , or|
|168||1||isomorphic to , which means we’ve missed the analogous group of order 120|
|240||4||now we have groups with prime factor 5, the one I had been expecting, i.e. is here, probably the prime modulo 4 is important here;|
|256||>0||I don’t feel like waiting this long, there are 56092 of this size|
|312||1||suspicion busted, is the unique instance, , and we missed a group at 264|
|456||1||prime factor 19 present, isomorphic to which is analogous to the unique group with the desired property of size 312|
|504||5||the structures of these groups are all very different (not just different semi-direct products etc.), whatever that may imply|
|512||>0||10494213 groups of this size might be a bit overkill to check|
|729||12||the first set of groups whose size doesn’t contain a prime factor 2!|
|768||>0||1090235 groups is too much|
|960||988||this is where interesting stuff will happen, one of these is a perfect group (see Perfect groups)|
|1024||>0||probably a shipload of examples, as there are 49487365422 groups of this size, but this is where the journey ends|
The calculations are still running, I will update this table when they are finished.
So much for that trivia. Another question that popped up was “Are there perfect groups that have this property?” A perfectly reasonable question, as perfect groups are the possible end points of derived series and are natural examples of derived groups. I wasn’t the first: Emanuel Kowalski asked himself What’s special with commutators in the Weyl group of ?. So the instance of size 960 was the first, what’s the second? There are two perfect groups of order 960 by the way, where the action of the semidirect product is different, but only one of them has the desired property.
A theorem of Isaacs, in his paper Commutators and the commutator subgroup gives a possible construction of examples. Unfortunately the smallest perfect group of this type is which is of size as we know from the construction of the wreath product. And I am not satisfied with the gap between 960 and a number somewhere around .
So we need a “fast” check of the property and apply it to the list of perfect groups. Using the characterization in Theorem 6.7 from On commutators in groups we find that the next perfect group, namely a non-split extension of with of size 1080 is already a perfect group satisfying the property.
A table containing these and further results (starting at the first hit):
|960||2||1||as discussed before|
|1080||1||1||as discussed before|
|1920||7||4||the tuples (1920,4), (1920,5), (1920,6) and (1920,7)|
|2160||1||1||just like 1920 this is twice the size of a previous occurrence|
|2448||1||0||this one is the simple group|
|2520||1||0||this one is the simple group|
|3420||1||0||this one is the simple group|
|3840||7||4||the tuples (3840,2), (3840,5), (3840,6) and (3840,7), it makes you wonder: is there a reason to renumber the perfect groups matching this behavior?|
|4080||1||0||this one is the simple group|
|4860||2||2||again there is an involved, as in most perfect groups but it seems to be an important constituent as the only group without an component is the group of size 2160 which contains an|
|5376||1||1||no , but a is the main factor|
|5616||1||0||this one is the simple group|
The class of perfect groups includes the non-abelian simple groups, but a conjecture by Ore (see Some remarks on commutators) which has been settled by Liebeck-O’Brien-Shalev-Tiep in The Ore conjecture states that for these groups every element is a commutator. So we can exclude the simple groups from our search space, as indicated in the table.
One final note, the property throughout this post boils down to the commutator length of a group not being equal to one. The commutator length is the minimal length of an element in the derived subgroup written as commutators of the original group. For more information see the already mentioned survey. So far no perfect groups of commutator length 3 (or more) have been found, if a group occurs in the table it has commutator length 2.
- I am aware of the uselessness of this little investigation, but I found the search interesting, and I learned some stuff about computational character theory, maybe someone else might be interested in these findings too;
- Currently the code is in no state to show, and as my exam is tomorrow I’d better study some more instead of polishing this. Besides, I should try Magma for these calculations, hopefully its implementation of character tables is faster…
- All groups in this post are considered to be finite. I don’t know which statements might be false for infinite groups.