11. Enhancements

We experimented with a new procedure to enumerate cosets. This has led to two frameworks and 9 strategies of which one framework and 7 strategies were evaluated further. It was found that the examples considered do not behave uniformly. It depends on the example whether there are best strategies or orderings and if there is a great variation of the number of cosets enumerated for the combinations evaluated or not. For most examples the enumeration process behaves quite differently depending largely on the combination of strategy and ordering. Further the idea of families of groups with respect to coset enumeration was evaluated.

Several enhancements are possible now. First of all the number of examples has to be expanded to give more information about the behaviour of this procedure compared to those already known for quite a while like HLT and Felsch type methods. To do that, the implementation was specialized in order to be able to compute larger examples as time and space requirements are very high for the more general implementation.

On the other hand there are other frameworks and other strategies possible which have to be evaluated. There are still two strategies left which allow more parameters. Further, examining the results of all available methods to enumerate cosets might lead to further improvements of the methods known or even to a brand-new method which is superior to the current ones.

Finally, the idea of families of groups has to be investigated more closely. The example considered showed a very varying behaviour with respect to coset enumeration using our procedures for the first members of the family while it seems to get stable for larger members. As the groups $PSL_2(11)$, $M_{11}^{(1)}$, and $Neu$ are not members of families of defining relations they can not be considered for testing this concept. The groups $E_1$ and $Cam(3)$ are such groups, but the next member of these families could not be computed with the resources available as they are already much harder. From the examples considered in this report, there remain the coxeter groups (see [2]) which form syntactic families and which have to be analyzed more closely.