As John says, not every n permits the existence of a non-Abelian group of order n. If n is a prime, then there is precisely one abstract group of order n and it is cyclic, hence Abelian.
Also as John says, the simplest example of a (barely) non-Abelian group is the dihedral group: the ways of fitting a cut-out regular polygon back in its hole, including flipping it over. Hence "dihedral" (=2-sided). I'd add that if n is odd, it enormously simplifies the possibilities. This is the notorious Feit-Thompson theorem, http://en.wikipedia.org/wiki/Feit%E2%80%93Thompson_theorem which in 1963 broke all records for the longest mathematical proof ever published. It states that all odd-order groups are solvable. "Solving" a group means breaking it down into its "composition series": http://en.wikipedia.org/wiki/Composition_series the components being called "simple" groups. The reverse process is called "group extension": http://en.wikipedia.org/wiki/Group_extension which offers the basis of a group-constructing algorithm. Simple groups include the cyclic groups of prime order, plus several series of groups whose structure is very far from simple. It's been a leading research problem for the past century to classify the finite simple groups, and in 2004 it was actually completed. http://en.wikipedia.org/wiki/List_of_finite_simple_groups In the 1960s it was one of the biggest international research programs in pure mathematics I knew of, drawing in some of the leading mathematical minds -- plus a few not-so-leading ones, like yours-truly :-) A finite group is called "solvable" only if its composition series contains none of these tough-nut non-cyclic, non-Abelian groups, but only prime-cyclics. I recommend this article, especially http://en.wikipedia.org/wiki/Group_extension#Extension_problem because it explains the problem deftly without plunging into the asphyxiating terminology of finite group theory. The title of my PhD dissertation was "the construction of finite factorisable groups" (1972) and for it I actually got free exclusive use of a large commercial mainframe (Burroughs B6500) to do a search for finite simple groups of a given order... or at least, a search for components called "elementary transcosets" which I needed to build up any finite group, even non-solvable ones, by a kind of generalized group extension called "transcoset theory". As with number theory, the difficulties only appear for astronomical values of n, the order of the target group G to be generated, far in excess of what even today's computers can handle. In 1972 Marshall Hall, jnr published a survey of all the simple groups up to n=1,000,000, and the number you'd have to consider in a short(ish) algorithm is manageable. http://en.wikipedia.org/wiki/List_of_finite_simple_groups#Non-cyclic_simple_groups_of_small_order If I was setting out today using J to find all possible G of given order n, I'd go right back to basics and work with the Cayley table of G (=multiplication table). Yes, really. Using heuristics of course to cut out the unwanted redundancy. Basically I'd choose an entry in Cayley(G) to be an integer, 0 to (n-1). But if A is a subgroup of G then the entries might become (<'A';p;q) where p and q are permutations of the rows and columns of Cayley(A) --this in effect builds Cayley(G) from the so-called right- and left-coset expansions of A in G. If you build G up by extending it by one prime-cyclic: Cp at a time, then Cayley(G) need only have (p^2) boxed entries. This is just my preferred approach. If you ask another group theorist, they'll more likely suggest representing the elements of G by the elements of Sym(n) (which contains as subgroups all possible groups G) -- ie as permutations, or as binary matrices (=permutation matrices) (as you say you've been trying) and working with so-called "group presentations". There's a conjecture that any (non-cyclic) simple group can be generated by [the right choice of] precisely two elements. I think it can be relied on for all practical values of n. If elements g and h generate G, then G can be represented as a set of equations in {g,h} -- this is called a "(group) presentation" of G. Deciding whether two given presentations are isomorphic, or even whether any given presentation is finite, are Turing-uncomputable problems (proven by Gregory Chaitin, if memory serves), so I've never thought of presentations as offering a viable computing technique. You may disagree, and discover they suit you well. Though once you've got your G, it's a neat way to define it. If I recall, all the new simple groups that have been discovered in the last 40 years have been published as group presentations. If you want a nice simple FAX to play around with theorem-proving techniques, then a (semi-)group presentation makes a splendid subject. I'd guess you'd need theorem-proving to generate the Cayley table of G from a presentation of it. Do you want me to find my dissertation and dust it off? IanClark On Fri, Dec 16, 2011 at 2:28 PM, Roger Hui <rogerhui.can...@gmail.com> wrote: > Does anyone know a way to generate a non-abelian group with order m? > > I can generate non-abelian groups, e.g. multiplication of non-singular > boolean matrices, or the group generated by a bunch of permutations, but I > need a group of size m when I am given the m. > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
