Permutation Groups Barry Dayton Northeastern Illinois University www.neiu.edu/ bhdayton June, 2002 Here is an elementary exposition of permutation groups making use of arrow diagrams to simplify matters. An explicit exposition of Lagrange’s Theorem in the case of a group acting on a set leads directly to an elementary exposition and proof of Burnside’s theorem. The prerequisites are simply a basic familiarity with the idea of a group with perhaps an acquaintance of the symmetry groups Cn;Dn and the modular arithmetic groups Zn. These notes were written for my Modern Algebra for Elementary School Teachers course, Summer 2002. 1 Permutation Groups A rearrangement of a set is a 1-1 correspondence from a set X to itself. In algebra, especially if the set is finite, these are called permutations. A little care must be used to distinguish our use of permutations with that in combinatorics, but, as you will see the concepts are quite related. Since the exact names of the members of the elements of X are unimportant, when X is finite we will usually assume that X = 1; 2; 3; : : : ; n for some positive integer n. And, except as noted below, we will assume X is finite.f We will useg the notation n(X) = n to denote the cardinality of this set. The easiest way to denote a permutation is to give its function table and/or its static arrow diagram. For example when n = 5 we might have: x f(x) 1 3 2 5 3 2 4 4 5 1 Figure 1: Static Arrow Diagram To combine permutations we view them as functions and compose them. That is given permuta- tions f; g then gf is the permutation given by gf(x) = g(f(x)). The easiest way to do this is with tables: 1 x f(x) x g(x) 1 3 1 4 2 5 2 2 3 2 3 1 4 4 4 5 5 1 5 3 The trick is to draw the static arrow diagrams and, rearranging, place them next to each other so that the middle columns match up, then ignore the middle columns. Figure 2: Compositon of Permutations Note that when we write gf it means that we do f first and g after. The order is important since combining permutations is not commutative in general. Clearly composition is associative and the do-nothing function ι(x) = x is a unity. To find the inverse, just reverse the arrows: Figure 3: Inverse Permutation Thus the set of permutations of X = 1; 2; 3; : : : ; n is a group. This group is generally denoted by f g Sn and called the symmetric group on n elements. The right hand column of a table representation 2 for a permutation is a permutation (re-arrangement) of the numbers 1; 2; : : : ; n and each such re- arrangement gives a different permutation. So by elementary combinatorics we see there are n! permutations of the set of n-elements. 2 Dynamic Representation and Cycle Decomposition It is useful to represent permutations by their dynamic arrow diagrams. Since the domain and range of these functions are the same we can list each element only once, with an arrow pointing towards the image of the element under the permutation. The definition of function requires that exactly one arrow leaves each element, but the fact that we actually have a 1-1 correspondence requires that there is exactly one arrow into each element. This requires that the diagram breaks into loops or cycles. Figure 4: Static and Dynamic Arrow Diagrams In this picture the diagram breaks into 3 cycles, one containing 1,3,5 another with 2,6 and the element 4 is alone in its own cycle since h(4) = 4. (4 is called a fixed point.) In general the dynamic arrow diagram is not convenient for composing permutations but in the special case we are composing a permution with itself it is very useful: for h2 go two steps around each cycle, for h3 three steps, etc. If we calculate powers of h of Figure 4 we get: 3 x h(x) x h2(x) x h3(x) x h4(x) x h5(x) x h6(x) 1 3 1 5 1 1 1 3 1 5 1 1 2 6 2 2 2 6 2 2 2 6 2 2 3 5 3 1 3 3 3 5 3 1 3 3 4 4 4 4 4 4 4 4 4 4 4 4 5 1 5 3 5 5 5 1 5 3 5 5 6 2 6 6 6 2 6 6 6 2 6 6 What we see is that if the length of a cycle divides an integer k then every element in that cycle remains fixed under hk. So in particular since the length of every cycle of h divides 6 then every element is fixed under h6 so, as seen by the table, h6 = ι the do-nothing. This works in general and we have the theorem: Theorem 1 Let m be the least common multiple of the lengths of all cycles of the permutation f = ι. Then f m = ι but f k = ι for 1 k < m. 6 6 ≤ We call the number m above the order of the permutation f. Exercise 1 Consider the permutations f; g of Figure 2. Find the dynamic arrow diagrams and the 2 3 1 orders of f; f ; f ; g; gf; fg and gfg− . For the permutation in Figure 4 we see that the different cycles work separately. Thus, as in Figure 5 below, we can write h as a product of cyclic permutations, permutations with only one non-trivial (length bigger than one) cycle. In the literature it is common to denote a cyclic permutation by a symbol such as (1 3 5) which denotes the cyclic permutation taking 1 to 3, 3 to 5 and 5 back to 1. Figure 5: Product of Disjoint Cycles In general, every permutation is, in this way, a product of disjoint cycles. Here the order in which we multiply does not matter since the cycles act on different elements of the set X. In texts on permutation groups you may see h represented by (1 3 5)(2 6). 4 3 Groups acting on Sets We can observe that many of our familiar groups remind us of permutation groups. For example since a symmetry of the square must take the vertices to vertices, each symmetry can be thought of as a permutation of the vertices. In this way D4 is isomorphic to a subgroup of S4 More generally we say the group G acts on the set X if each element g G can be represented by a permutation g(x) of X in such a way that 2 (i) For g; h G the combination gh is represented by the composition g(h(x)) 2 (ii) The unity in G is represented by ι, the do-nothing. We do not require here that different elements g; h of G are represented by different permuta- tions g(x); h(x), however if this does happen then G is isomorphic to a subgroup of the group of permutations on X. Figure 6: The Square Consider again the symmetry group D4 of the square (Figure 6). We represent r by the cycle (1 2 j k 3 4) and m as the product of cycles (1 2)(3 4). Then a typical element of D4 is of the form r m where j = 0; 1; 2 or 3 and k = 0 or 1. Then rjmk should go to (1 2 3 4)j(1 2)k(3 4)k The reader should generate the multiplication table for these permutations herself and check the the resulting subgroup of S4 is, in fact, isomorpic to D4. Given a group G we can define for each a G a function a(x) from G to itself by a(x) = ax for all x, where ax is multiplication in the group.2 The existence of inverses in the group implies that this function has an inverse function and therefore must be a permutation of G. Laws (i) and (ii) hold so G is acting on itself, here X = G. Finally cancellation again says different elements of G act differently so we have shown: Cayley’s Theorem Every group is isomorphic to a subgroup of a permutation group. 5 In the special case of a finite group G Cayley’s theorem says that G is isomorphic to a subgroup of Sn where n = n(G). Unfortunately for group theorists this does not help much in understanding finite groups. But in a sense, the symmetric groups are the most general groups. Let G act on X. We say an element a X is a fixed point, or an invariant for group element f if f(w) = w. For a given set element2w the set of all invariant group elements f with w as an invariant is called the isotropy subgroup, in symbols I(w) = f G f(w) = w . On the other hand the orbit of w is the set of all set elements which are imagesf 2 of jw under someg permutation in G, in symbols orb(w) = x X f(w) = x for some f G . The French mathematician Lagrange noticed in some specificf 2 examplesj that there was a very2 strikingg relationship between the sizes of these sets and n(G). Lets look at an example. Figure 7: Orbits We let G be the subgroup of S7 consisting of the cycle R = (1 2 3 4), the cycle r = (5 6 7) and all powers and products of those permutations.