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Pacific Journal of Mathematics ON THE LATTICE OF NORMAL SUBGROUPS OF A DIRECT PRODUCT M ICHAEL D OUGLAS M ILLER Vol. 60, No. 2 October 1975 PACIFIC JOURNAL OF MATHEMATICS Vol. 60, No. 2, 1975 ON THE LATTICE OF NORMAL SUBGROUPS OF A DIRECT PRODUCT MICHAEL D . MILLER Suzuki has determined that if G is a direct product G = Π^jG, of groups Gt ^ 1, then the lattice L ( G ) of subgroups of G is the direct product of the lattices L(Gi) if and only if the order of any element in G, is finite and relatively prime to the order of any element in G,(ίV /). An exercise in Zassenhaus' The Theory of Groups asks the reader to prove an analogous result for the lattice of normal subgroups. In §1, we derive this result for the case of the direct product of two groups. (The generalization to the direct product of any finite number of groups is straightforward.) In §2, we use results obtained in §1 to study in detail the normal subgroup lattice of the direct product of finitely many symmetric groups. 1. The lattice of normal subgroups. If Gλ and G2 are groups, we denote elements of the direct product Gx x G2 by ordered pairs (a, />), a E Gu b E G2. If A and B are subgroups of a group G, we ι ι define [A,B] = (aba~ b~ \a E A, b E B), and note that if A < G, then [A, B]<lA. We let pi and p 2 denote the first and second projection maps on Gλ x G2, andfinally,we denote by o(g) the order of the element g If N is a subgroup of Gx x G2, we put Nγ = p\{N) and JV2 = p2(N). Thus Nι is a subgroup of G, , called the ith projection of N. Furthermore, if N < Gx x G2, then Ns < Gf, LEMMA 1. IfN<Gιx G2, then N D [Gu JV,] x [G2, N 2 ]. Proof. Let a E Nλ. Then there exists y E N 2 such that (a,y)E N. Thus ( α ^ y - ^ E N , and since N<GιxG2, (g, l)(α, yXg'1,1) = ι (g^gΛ y) e N. It follows that (gag-\ y){a'\ y~ ) - (gag^a~\ 1) E ΛΓ, so N D [Gu NJ X {1}. Similarly, N D {1} x [G2, N 2 ], completing the proof. The following lemma, whose proof is immediate, will be used in the discussion that follows. 2. Let G be a group, H < G. such that [G,H]CL CH is normal in G. LEMMA Then any subgroup L of G Since [GλxG2,A x J5] = [Gu A] x [G 2 ,B] whenever ACGUBC G2, if N < Gi x G 2 with projections Nx and N 2 , then any subgroup of GiXG 2 lying between [GuNi]x[G2,N2] and Nx x N2 is normal in 153 154 MICHAEL D. MILLER Gί x G2. Moreover, as [Nh Nt] C [Gh Λζ-], we see that Q = Ni /[Gf , Nj] is abelian, as is d x C2 = (ΛΓi x N2)/[GU Nt] x [G2, N 2 ]. There is thus a 1-1 correspondence φ between subgroups of d x C2 and subgroups of G! x G2 lying between [Gl9 Nt] x [G2, ΛΓ2] and N2 x JV2. DEFINITION. A normal subgroup S of Gx x G2 is called Gi - G 2 decomposable if 5 = Sι x S2, Si <l G 1? S2 <l G2. It is easy to see that a subgroup // of d x C2 is Ci - C2 decomposable if and only if φ(H) is Gx- G2 decomposable. Furthermore, if B C C i X C2, then the ith projection of B is Q if and only if the ith projection of φ(B) is N,. Assuming we can determine the subgroup lattice structure of arbitrary abelian groups, we now have a systematic way of describing the normal subgroups of Gλ x G 2 in terms of those of Gx and G2. Namely, choose Si <J Gi, S2 < G2 and consider the subgroups M of the abelian group St/lGuS^x S2/[G2,S2] with the property that M, = SJ[Gh S,]. To each such M, there corresponds a normal subgroup N of G! x G2 with [Gu Sι] x [G2, S J C N C ^ x S2 and N, = S, . As S t and S2 run through the normal subgroups of G1 and G2 respectively, we obtain each normal subgroup of Gλ x G 2 exactly once. It is, of course, not always easy to determine all subgroups of a given abelian group. For finite groups, however, Suzuki's result shows that it suffices to consider the case of abelian p-groups. Let Gλ = S3 (symmetric group on 3 letters) G2 = Z2 (cyclic group of order 2) We calculate the following: EXAMPLE. Si s2 [Gu S,] [G2, S2] c, c2 V 1 1 1 1 1 1 1 1 z2 1 1 1 1 z3 z2 1 1 1 1 Z3 1 1 1 z3 z3 1 z2 z2 z2 1 1 z2 2 z3 z3 s3 S3 1 z2 1 z2 1 Here Q; = Si/[Gh S, ], and ^ denotes the number of subgroups M C d x C2 with Mi = C From this, we see that S3 x Z2 has seven normal subgroups, all of which are S3 - Z2 decomposable, except for one of order six. ON THE LATTICE OF NORMAL SUBGROUPS 155 We now determine the condition for every normal subgroup of Gx x G2 to be Gι - G2 decomposable. Recall that a group G is called perfect if G = G'. We will say that G is super-perfect if [G, H] = H for all H <3 G. 1. Let Gx and G2 be groups. Then every normal subgroup of Gι x G 2 is Gι - G2 decomposable if and only if either (i) at least one of Gi and G2 is super-perfect, or (ii) for all Sx < Gx, S2 < G2, the elements of SJ[GUSX] have orders relatively prime to those of S2/[G2,S2]. (In particular, these orders must be finite.) THEOREM Proof ( <= ) Suppose N<Gxx G2 is not Gx- G2 decomposable. Then the subgroup φ(N) of NX/[GUNX] x N2/[G2,N2] = Cx x C2 is not Cx~ C2 decomposable. If G, is super-perfect, then Q? = 1, a contradiction. Otherwise (ii) holds, and we have a contradiction to Suzuki's result [2]. ( φ ) Let 5, <3 G,. By hypothesis, every normal subgroup NCGxxG2 with [Gx, Sx] x [G2, S J C J V C ^ x S2 is Gi - G2 decomposable, and therefore all subgroups of SX/[GX, Sx] x S2/[G2, S2] = Ci x C2 are Cx - C2 decomposable. If Cx has elements of infinite order, then G2 must be super-perfect. For if not, there is a normal subgroup H of G2 such that D = H/[G2, H] ^ 1. By Suzuki's result, d x D contains a subgroup which is not Cx - D decomposable, a contradiction. Simililarly, if Gλ is not super-perfect, then C2 must be a torsion group. Finally, if neither Gλ nor G2 is super-perfect, then the order of any element in Cλ must be relatively prime to the order of any element in C2, for if not, by Suzuki's result, there would be a subgroup of CΊ x C2 which is not Ci - C2 decomposable. COROLLARY 1. Every normal subgroup of G x G is G - G decomposable if and only if G is super-perfect COROLLARY 2. // Gλ and G 2 are torsion groups, and the order of any element in Gλ is relatively prime to the order of any element in G2, then every normal subgroup of Gx x G2 is Gx - G2 decomposable. DEFINITION. A torsion group G is called quasi-nilpotent if for every prime p with p = o(a) for some a E G, there exists H <3G such that H/[G,H] has an element of order p. It is easy to see that every nilpotent torsion group is quasinilpotent. The first example of a quasi-nilpotent group which is not nilpotent is the group S3 x Z3, of order 18. The first indecomposable example is SL(2,3) of order 24. 156 MICHAEL D. MILLER We now state a partial converse to Corollary 2: COROLLARY 3. // Gλ and G2 are quasi-nilpotent groups such that every normal subgroup of Gλ x G2 is Gx - G2 decomposable, then the order of any element in Gλ is relatively prime to the order of any element in G 2 . Theorem 1 can be easily generalized to yield the following: THEOREM. 2. Let G = Πf=1 G, . Then every normal subgroup N of G is a direct product N = Πf=1JVi of normal subgroups N< of Gt if and only if either (i) at most one of the G, is not super-perfect, or (ii) whenever Ht < G, and Hj < G ; (ιV/), the order of any element in Hi/[GhHi] is relatively prime to the order of any element in Hjl[Gj9Hj\. (In particular, these orders must be finite.) The proof is identical in nature to that of Theorem 1, and will therefore be omitted. Instead of studying the lattice of normal subgroups, one can look at other systems of subgroups which form a lattice. For example, one could ask when every characteristic (resp. fully invariant) subgroup of nf=1Gi is a direct product of characteristic (resp. fully invariant) subgroups of the individual G, . These problems appear to be substantially more difficult than the one treated in this section. 2. Direct products of symmetric groups. We begin with two definitions. If G is a group and H is any subgroup containing G', then H is called a CC-subgroup of G. All CC-subgroups are therefore normal. Secondly, if G = IΊf=1G/ and p, is the projection on the /th factor, then an automorphism φ of G is called rigid if φ(pi(G)) = ρt(G) for all i. The group of rigid automorphisms of G is thus isomorphic to Πf=1 Aut(G, ). Now let (Sn)k be the direct product of k copies of the symmetric group 5n, where n > 4. (The results of this section are not in general true for n ^ 4, although analogous results may be obtained by treating each case separately.) We wish to determine all normal subgroups of (Sn)k. Forfc= 1, there are exactly three: 1, An9 and Sn. Suppose the normal subgroups of (Sn)r for r<k have been determined. Then if N<(Sn)k, we may assume that N is not contained in a product of fewer than k copies of Sn. By the simplicity of A n , N D ( A n ) k and so JV is a CC-subgroup. Now (Sn)k/(An)k is an elementary abelian 2-group, which may be considered as the vector space P (over Z 2 ) of subsets of the set K = {1,2, , fc}, where addition is defined by symmetric difference. There is therefore a 1 - 1 correspondence σ between CC- ON THE LATTICE OF NORMAL SUBGROUPS 157 subgroups of (Sn)k and subspaces of P. We proceed to show that this correspondence can be made canonical. If N is a CC-subgroup of (Sn)*, let σ(N) be the subspace of P spanned by the set of all U CK such that ΐlieuXi E An for all (JCI, , x k ) £ N. Conversely, if S is a subspace of P, let N be that CCk subgroup of (Sn) consisting of all elements (xux2, *,**) such that UieRXi £ Λ n for all RES. It is easily verified that σ(N) = 5, and that σ is a Galois correspondence, i.e., it is 1 - 1 and reverses inclusion. For example, (Snf/(Anf is isomorphic to the Klein group Z 2 x Z 2 , which has 5 subgroups (subspaces). There are therefore 5 CCsubgroups <rf (5 n ) 2 viz., (Λ π ) 2 , An x Sn, Sn x An, {(JC 1 ,X 2 )|X 1 X 2 6 An}, and (S n ) 2 . If we add to these the subgroups 1 x 1,1 x An, An x 1,1 x Sn, and Sn x 1, we find that there is a total of 10 normal subgroups in (5 n ) 2 . Now eue2,'',ek where £,={/} form a basis of P. Under σ,ex corresponds to the subgroup of all (x 1 ,x 2 , ,xk)E(Sn)k such that xf is even. We define a coordinate plane to be a subspace of P spanned by some collection of the eh and call it proper if it has dimension < fc. It is not hard to see that a CC-subgroup N of (Sn)k is a nontrivial direct k product of two normal subgroups of (Sn) if and only if σ(N) is the direct sum of two subspaces of P, contained respectively in complementary proper coordinate planes. We now recall the following well-known facts: (i) For n > 4, n ^ 6, Aut Sn = Aut A n = Sn. Moreover, Aut S6 = Aut A 6 and [Aut S 6 : Inn S6] = 2. (ii) (Mathewson [1]) For n > 4, Aut (Sn )k = Aut (AB )k = In words, eyery automorphism of Sn(n > 4 , ny^β) is inner, while every automorphism of (Sn)k(n > 4 , n^ 6) is the product of an inner automorphism and an automorphism which permutes the k factors. For k n > 4, the automorphism group of An (resp. (An) ) is the same as that of k Sn (resp. (Sn) ). k 3. Let N be a CC-subgroup of (Sn) . Then every k automorphism of N is induced by an automorphism of (Sn) . THEOREM Proof Let θ G Aut N. By the above, the action of θ on (Anf is that of the product of a rigid automorphism of (A n )k and an automorphism which permutes the k factors of (An)k. Multiplying θ by a rigid automorphism of (Sn)k (itself an automorphism of JV), we may assume that the action of θ on (An)k is simply a permutation TΓ of the k factors. Let xEN with θ(x) = y and eE(An)k with θ{e) = e". Since -1 x x ex E(A n )*, we have Q(x~ ex)~ x'^e^x^. But also β(jc"1ejc) = π 2 y^e'y. As e is arbitrary in (A n )\ x y~ is in the centralizer of (An)fc, which is trivial, so y = x π . Thus every automorphism of N is the 158 MICHAEL D. MILLER k product of a rigid automorphism of (Sn) and an automorphism which permutes the k factors. The theorem follows. In general, not all automorphisms of (Sn)k actually restrict to automorphisms of a given CC- subgroup JV, since not all permutations of the k factors leave N invariant. If S = σ(N) is the corresponding subspace of P, let Γ denote the group of permutation matrices (with respect to the basis eue29 —,ek) which leave 5 invariant. Then = (AutSn)fc xs Γ. The characteristic subgroups of (Sn)k are: and Γ2, where Tλ = {(xu x2, , Jck)|Πf=1x,, e An}9 and , Xk)\ΐlι<pCiXj E An for all /,/}. (Note that Tx = T2 in case THEOREM 4. l,(Sn)\(An)\Tu T2 = {(xu *2, fc=2.) Proof It is clear that except for 1, any characteristic subgroup of k k (Sn) contains (An) , for otherwise it would be contained in a direct product of fewer than k copies of Sn, and hence not be characteristic. In terms of P, we must show that the only subspaces 5 invariant under all permutations of the coordinates are 0 , P9 the 1-dimensional subspace Vx spanned by eλ + e2λ + ek, and the (k — l)-dimensional subspace V2 spanned by all eι + er That these are all invariant is immediate. Assume now that 5 is invariant, and suppose that axeλ + + arer H + a5es + 4- akek G 5, where ar + as ^ 0 for some choice of r and s (otherwise S = 0 or S = Vi). By invariance, α^i + + ares + + aser 4+ akek G 5, and adding gives (ar + as)er + (αr + as)es G S, so that er + es G 5. Again by invariance, we conclude that e{ + e^S for all i, /, so either S = P or S = V2. The question of determining exactly which groups can arise as the group Γ of "admissible" permutation matrices for a given CC-subgroup N will be dealt with in a future paper. REFERENCES 1. L. C. Mathewson, Theorems on the groups of isomorphisms of certain groups, Amer. J. Math., 38 (1916), 19-44. 2. M. Suzuki, Structure of a Group and the Structure of its Lattice of Subgroups, Springer-Verlag, 1967. 3. H. Zassenhaus, The Theory of Groups, Chelsea, 1958. Received September 3, 1974, and in revised form October 14, 1974. UNIVERSITY OF CALIFORNIA, BERKELEY PACIFIC JOURNAL OF MATHEMATICS EDITORS RICHARD ARENS (Managing Editor) J. DUGUNDJI University of California Los Angeles, California 90024 Department of Mathematics University of Southern California Los Angeles, California 90007 R. A. BEAUMONT D . GlLBARG AND J. MlLGRAM Stanford University Stanford, California 94305 University of Washington Seattle, Washington 98105 ASSOCIATE EDITORS E F. BECKENBACH B. H. NEUMANN R WOLF K. 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Subscriptions, orders for back numbers, and changes of address should be sent to Pacific Journal of Mathematics, 103 Highland Boulevard, Berkeley, California, 94708. PUBLISHED BY PACIFIC JOURNAL OF MATHEMATICS, A NON-PROFIT CORPORATION Printed at Jerusalem Academic Press, POB 2390, Jerusalem, Israel. Copyright © 1975 Pacific Journal of Mathematics All Rights Reserved Pacific Journal of Mathematics Vol. 60, No. 2 October, 1975 Waleed A. Al-Salam and A. Verma, A fractional Leibniz q-formula . . . . . . . . . . . . . Robert A. Bekes, Algebraically irreducible representations of L 1 (G). . . . . . . . . . . . Thomas Theodore Bowman, Construction functors for topological semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stephen LaVern Campbell, Operator-valued inner functions analytic on the closed disc. II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Leonard Eliezer Dor and Edward Wilfred Odell, Jr., Monotone bases in L p . . . . . . Yukiyoshi Ebihara, Mitsuhiro Nakao and Tokumori Nanbu, On the existence of global classical solution of initial-boundary value problem for cmu − u 3 = f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Gordon, Unconditional Schauder decompositions of normed ideals of operators between some l p -spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gary Grefsrud, Oscillatory properties of solutions of certain nth order functional differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irvin Roy Hentzel, Generalized right alternative rings . . . . . . . . . . . . . . . . . . . . . . . . . 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