Guralnick theorem

Exercise. Prove the following result:

Theorem (Guralnick). There is a group $G$ of order $\leq200$ such that $[G,G]\ne\lbrace [x,y]:x,y\in G\rbrace$ if and only if $n\in\lbrace 96,128,144, 162, 168, 192\rbrace$.

The theorem appeared in [Guralnick, R. Commutators and commutator subgroups. Adv. in Math. 45 (1982), no. 3, 319–330].

Solution. To simplify a bit the presentation, we first define a function that returns the set of commutators of a group:

> SetOfCommutators := function(G)
function> return { (x,y) : x,y in G };
function> end function;

Now we use this function. We send the parameter Warning := false to the function SmallGroups to avoid the warning messages. Otherwise, Magma warns us that the requested calculation involves a huge number of groups:

> time { #G : G in SmallGroups([1..200]:Warning:=false) | 
> not #DerivedSubgroup(G) eq #SetOfCommutators(G) };
{ 96, 128, 144, 162, 168, 192 }
Time: 34.360

The calculation took 34 seconds.