Hi Nilo, As you have noticed, the 'shape' of a group, i.e., what you get from StructureDescription(), does not determine the isomorphism type of the group.
There are two split extensions of (4 x 2) by 2, the action of the outer 2 on the normal 4x2 is different. With Ken's notation we have ( <x> x <y> ) : <z>. Now, 4x2 has two cyclic subgroups of order 4, one generated by x that we can see clearly, the other generated by xy. In case [16,3] z acts to swap these. 4x2 has three cyclic subgroups of order 2, generators are x^2, y and x^2y. In case [16,3] z acts to swap <y> with <x^2 y>. Notice that this fixes the two subgroups of order 4 You can't swap any other pair of order 2 subgroups. For example, y -> x^2 -> xyxy = x^2 -> x^2 -> ... which is nonsense. It is also impossible to do both of these swaps at once: The first requires you to swap x with xy, the second forces you to swap x with x^3y. Therefore these are the only two groups of shape (4x2):2. I seem to remember that "Groups for Undergraduates" by J. Moody determines all of the groups of order up <something>. I expect there are many other references. Richard. > Hi Nilo, > Both groups are semidirect products of a normal subgroup isomorphic to > C4 x C2 with a subgroup of order 2. > > (More explicitly, according to some notes of mine, group [16,3] is > generated by elements x, y, z where x has order 4, y and z have order > 2, x and y commute (thus <x,y> = C4 x C2), y and z commute and zxz=xy. > Group [16,13] is generated by x, y, z with orders 4, 2, 2, respectively > where xy=yx, xz=zx, zyz=x^2y.) > > ken > --- > On Feb 10, 2006, at 4:11 AM, Nilo de Roock wrote: > >> Hello GAP Forum, >> >> Could someone please explain why AllGroups(16)[3] and >> AllGroups(16)[13] both return "(C4 x C2) : C2" on the function >> StructureDescription? >> >> Thanks in advance, >> nilo >> >> _______________________________________________ >> Forum mailing list >> [email protected] >> http://mail.gap-system.org/mailman/listinfo/forum >> >> > --- > Ken W. Smith, Professor of Mathematics, Central Michigan University > 989-854-0185 (Cell) > http://www.cst.cmich.edu/users/smith1kw > Address for 2005-06: > 22 Chase Gayton Terrace, Apt 1518 > Richmond, VA 23238-6526 > > _______________________________________________ > Forum mailing list > [email protected] > http://mail.gap-system.org/mailman/listinfo/forum _______________________________________________ Forum mailing list [email protected] http://mail.gap-system.org/mailman/listinfo/forum
