TRANSPOSITIONS IN FINITE SYMMETRIC GROUPS. Lynnette Gilmore Peter Lorimer*

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1 TRANSPOSITIONS IN FINITE SYMMETRIC GROUPS Lynnette Gilmore Peter Lorimer* (received 17 October 1972; revised 10 September 1973) The symmetric group S^ on a set E is the group of all bisections or permutations E E with binary operation composition of functions. If a, 8 are two different members of E the permutation which interchanges a and 8 and leaves every other element of E fixed is denoted by (a8) and is called a transposition. The transpositions of S^ form a conjugacy class of S^. We will denote this class by C. The study of transpositions has played an important role in the theory of symmetric groups. Most importantly, S is generated by C, but also a study of C leads to the classical result that is complete (that is, having trivial centre and no outer automorphisms) when E is a finite set of cardinal number not 6. Here we concentrate on the latter type of result and prove Theorem 1 which seems to be the furthest it is possible to go in this direction. As a corollary to Theorem 1 we prove the above result about completeness provided E has cardinal number different from 4 or 6. Finally we mention the relationship between Theorem 1 and a result of G.Sabidussi on the automorphism groups of derived graphs. THEOREM 1 Let C be the class of transpositions in the syrmetric group S on a finite set E and let V be a subset of C with the property: (A) if faoijjj (aa ), (aa^) are three different transpositions of V, then V contains a transposition (&y) with 8 {a,«i,0^,0^}, y {o,o1,a2,a3>. * While this work was completed the second author was partially supported by National Research Council grant A7544 at the University of Victoria, British Columbia. Math. Chronicle 3(1974),

2 If f : V -* C is an injection with the property: (B) For all xsy V, xy = yx ~ ffa)f(y) = f(y)f(.x), then there is a member W of S^ with the property: For every x i V, f(x) = wccw~^. Before proving Theorem 1 we prove two lemmas which contain the essential facts about transpositions that will be used in the proof. Lemma 1 If x = (ab) and y = (y6) are transpositions then xy t yx if and only if the set <S) contains three members. Proof Since a ± 6 and y ± 5, the set {a,b,y><5} contains at least two members, If {a,6,y,6} has only two members then (ab) = (y5) and (ab) (y<5) = (y6)(ag). If {a,8,y,6} has four members we have again (ab)(ys) = (y5)(ag). If {a,b,y><5} has three members we may suppose without loss of generality that a = y and a,6,6 are all different. Then (ab)(y5) = (ab<5) t (a6b) = Cy^D (otb)- Lemma 2 If ^jbjy are all different, then every transposition commutes with at least one of the transpositions (ab)., (By), (ya)* Proof If (6e) is a transposition, not all of the sets {a,b,<s,e}, (B,y,6,e} {y,a,6,e} can contain three members. Thus the result follows from Lemma 1. We now establish Theorem 1. Suppose that E, V and f are as in the statement of Theorem 1. For each a f I denote by <p(a) the cardinal number of the set of transpositions of V which are of the form (aa^. If <t>(a) > 1 we will show first that there is a member aw of E with the property that if (aa^ V then /(aa^) = (a^b^ for some Bj. aw will be 90

3 uniquely defined if (f>(a) > 1 but may possibly be one of two members of Z if <J>( 0 = 1. Suppose that <f>(a) > 2 and that (aa^, (aa2) are two members of V. As (aa^ and (aa2) do not commute, neither do /(ao^) and /(aa2). Hence /(ao^) = (33^ and /(aa2) = CBBg) for some 6,3j,B2 in Z. We define aw = B- We show now that a is well defined. This is clear if <f>(co = 2. Suppose <K 0-3 and (aa^ is a third member of V. As (aa3) commutes with neither (aoij) nor (aa2), f(aa^) commutes with neither (BBj) nor (832). The only possibilities are that /(aa ) =(33) or /(aa ) = (330) for some 3_ Z. Suppose the former is true. By hypothesis 04), V contains a transposition (y6) with y {a,a1,a2,a3> and 6 {a,aj,a2,a3>. Suppose y = a. Then (y<$) commutes with none of (ao^), (aa2), (aa3), and so /(y6) commutes with none of (33^, (332),(BjB^, which is impossible. Suppose y ^ a. Without loss of generality we may suppose y = a^. Then (y6) does not commute with (ac^) and so /(y6) does not commute with (33^- Hence /(Y<$) = (33 ) or /(y6) = (3^^) for some 3^ Z. Suppose the former. As f is an injection 3, ^ 3 and so (33 ) does not commute with C832) = /(aa2). Hence (y6) does not commute with (aa2) and so the set {a,a2,y,6} contains three members. But y = a {: {a,a2> and so 6 {a,a }, which is not true. Alternatively /(y6) = (3-^3^) for some 3 Z. Again we have (3,3. ) t (3.3.) and so (3,8, ) does not commute with = /(aa ) and so {a,a3,y,6} contains three members. As Y = ctj {a,a3> we obtain 6 {a,a3}, which is again a contradiction. This proves that we cannot have fcaa^ = C3j3^) and so /( aa^ = (883) for some 3^ ^ I. We have shown that if <J>(a) - 2 there is a uniquely defined member 3 = aw of Z with the property that if (a ) V then /(a ) = (Bn) for some n Z. Suppose that <j>(a) = 1 and (a ) V, /(a ) = (3n) If < K O - 2 we have already defined E,W as 3 or r\. We define aw as the other. Clearly aw is uniquely defined. If <KO = 1 w define aw arbitrarily as one of 3, r, and E,W as the other. In this case aw is not uniquely defined. 91

4 In any case we have shown that if 4>( 0-1 there is a member aw of Z such that if (a B ) V 3 j (a B ) = (awy) for some y Z. Clearly / O B ) = o V ). Now put Z^ = {a Z: <J>(a) - 1}- The mapping a -> aw is clearly an injection Z - Z and so there is a bijection w : -* Z with the property that w(a) = aw for each a Z.. the above. Suppose x = (a6) V. Then we have wxw~ 1 = (aw pw ) = f(x) by This proves Theorem 1. A consequence of Theorem 1 is the following classical result. THEOREM 2 If n t 4 or 6 and Z is a finite set with n members S i s complete. then Proof The transpositions C of form a set with the property that if x3 y C the order of the product xy is 1, 2 or 3. If C x is another class of involutions of S ^ it can easily be shown that, because n t 4 or 6, Ccontains two members whose product has order greater than 3. Hence any automorphism of S ^ must leave the class C fixed. Let / be an automorphism of S^. As /(C) = C if follows from Theorem 1 that S contains a member w with the property f(x) = w x w~l for every x C. As the set C generates S we then have fix) = w X w-1 for every x 6 S. Hence / is an inner automorphism. 92

5 Finally we will remark on the relation between Theorem 1 result of G. Sabidussi on derived graphs. and a Let G be a graph which we will take as a set V of vertices and a set E of edges where each edge will be a 2-element subset of V. From G we can produce another graph D called the derived graph of G. The vertices of D are the members of E and two members of E are joined by a line in D if and only if they meet at a point in G. Hence if {a,3> and {y,5> lie in E they are joined by an edge in D if and only if the set {a,3,y,6} contains three members. This should be compared with Lemma 1 above. Sabidussi's result (see [l]) may be stated as : except for a finite number of cases (all of which are subgraphs of the complete graph on 4 vertices), if / is a graph automorphism of D then there is a graph automorphism w of G such that /{a,6) = {zj(a),w(g)} for each {a,g} E. This result corresponds to Theorem 1 and is proved in a very similar way. From this we can deduce that, except in a finite number of cases (all of which are subgraphs of the complete graph on 4 vertices), the automorphism group of a graph is isomorphic to the automorphism group of its derived graph. This corresponds to Theorem 2. REFERENCE 1. G. Sabidussi, Graph Derivativess Math. Z. 76(1961), University of Auckland 93

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