2009/5/5 Gabriel Bartolini <ga...@mai.liu.se>: > Hi, > > I've been searching around but haven't found if it is possible to list all > groups of order N generated by elements of orders m1 to mr, for example; > all groups of order 32 generated by two elements x1, x2 of order 8 such that > order(x1*x2)=2. > > Any suggestions would be appreciated. > > Regards, > Gabriel
Hi Gabriel, I can think of two approaches, one of which won't work for this example and one of which will. The first approach would be to form the group with presentation <x1, x2 | x1^8 = x2^8 = (x1*x2)^2 = 1>, and see if it's finite. In this case it isn't, so we'd continue with the next approach; if it is, say of size N, you can see if it's possible to find all its normal subgroups of order N/32. Then mod those out and see in which cases the images of x1 and x2 still have the properties that you are looking for (instead of having an order that divides 8, for example). For the second approach, first find all groups of order 32 with the command AllSmallGroups(32). Then for each group G, find the conjugacy classes C1, C2, ..., of elements of order 8. Now find all representative pairs of elements from those conjugacy classes: for each class, take one representative element g1 as the first item in the pair; compute the stabilizer S of g1 in the automorphism group of G; then form a pair [g1, g2] for representatives g2 from all S-orbits of C1, C2, ... . Finally, for each such pair [g1, g2], check the order of g1 * g2 and whether g1 and g2 generate all of G. Of course, once you've found a pair that works, you don't need to check any of the other pairs for that group. HTH, Erik Postma. _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
