Lattice Automorphisms Citation Avann, Sherwin Parker (1940) Lattice Automorphisms. Master's thesis, California Institute of Technology. doi:10.7907/1fas-2w25. https://resolver.caltech.edu/CaltechTHESIS:12072024-130659977 Abstract No abstract. Item Type: Thesis (Master's thesis) Subject Keywords: (Mathematics) Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): Ward, Morgan Thesis Committee: Unknown, Unknown Defense Date: 1 January 1940 Record Number: CaltechTHESIS:12072024-130659977 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:12072024-130659977 DOI: 10.7907/1fas-2w25 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 16906 Collection: CaltechTHESIS Deposited By: Benjamin Perez Deposited On: 11 Dec 2024 18:14 Last Modified: 11 Dec 2024 18:30 Thesis Files PDF - Final Version See Usage Policy. 8MB Repository Staff Only: item control page

LATTICE AUTOMORPHISMS Thesis by Sherwin P. Avann In Partial Fulfillment of the Requirements for the Degree of Master of Science California Institute of Technology Pasadena, California TAJ3LE OF CONTENTS I Introduction Section Page 1. SUilllnary . . . . . 2. Nota,tion and Terminology • . . . . .. . ... . ........ 1 2 II The Group of Automorphisms of a Lattice 3. Definition ................ 3 4. Fundamental Properties of Automorphisms .. 4 5. Permutation Properties . . . . . . . . . . 6 6. Invariant Sublattices and Subgroups . . . . . . 8 7. Point Lattices with the Symmetric Group . . . . 10 8. A Notation for Distributive Lattices .. 15 .. 17 . . . . .... . 9. Direct Product of Linear Lattices. 10. Lattice with a Given Group 11. Two Examples . . . . . . . ........ 19 24 III Supplementary Results . . . . . . .. . . . ... . 13. Conclusion . • . . . . . . . . . . . . . . . . 30 References. . . . . . . . . . . . . . . . . . • 31 12. Modular Lattices 27 I Introduction 1. Summa ry. The s~bj e ct of lattice automorphisms ha s been introducecl by G. Eirkhoff in his paper 11 011 the Structure of Abstract Algebre,s 11 (Eirkhoff furth er. [3) ) . In this paper t h e subject ha s been pursuect Among the resu l ts are a nece s sary a nd sufficient condition that a p oint lattice with n points ha ve the symmetr:i.c group on n letters a s its group of automorphisms, a nd. the construction of a lattice with a given arbitrary abelian automorphi sm group. A few r e sults are added which are not directly connected with the stua.y of automorphisms . Of the se is a test for modula.rity of a lattice in terms of bonds (coverings) which is u seful. For hi s many helpful suggestion s during the course of this investigation I am indebted to Professor Morgan Ward. 2 2. Notation ancl terminolo&t. We shall generally d.enote a lattice by a script J:.. and its elements by capitals means A A contc1.ins we write .I:,,. is an element of ] A= E. A ;> :B (A < A C). A :> E (A We write (A conta.ined in C). A, B, C, X, Y. A ? When and :S covers B (A is covered. by C) A I'\ B and < C) AG~ for A C J3, is written (A, E) indicate the join and meet Let ~ represent a set of elements of n write A ; we Ai , Az' cl.. 4 A.1, = Ai I'\ Az /\ ' n i n ) J = At:, and b. 1n = b. A.1, = (Ai , Az, /\ An ' An i and M = t. i.,, will repr e sent the join and. meet respectively of all respectively, of A and :B . l\~= ... elements of the l attice. N C A for every lir E £, We call . a node if The elements covering A E J::... points and those covered by J dual :points. lattices will be represented. by d.3 and. A C N • or M are ca lled Dedekind or modul a.r distributive lattices by (!.. Sublattices 1,rill be in<Hcated by 1<.. . The automorphism group will be denoted by ,h and a subgroup by The symmetric grou.p on 1, letters is denoted by J need.ea_, 11 . ?+. . 1, /W New note,tions wil l be introducecl when II The Group of Automorphisms of a Lattice 3. Definition. An automorphism of a lattice£ is a one-to-one correspondence between the elements of£, and themselves which is preserved under the two fundamental operations of join and meet in We denote the automorphisms of If a to A by Aa and write: A > • 11 of an i, unage A. a.: a .r:A.~ by small Greek letters. £ , we denote the element corresponding is an automorphism of call , i,r,e £_ £. By definition: fo1• all A = (A = (Aa,, Ba,) B)a £., . By an e2,sy induction ,;e get: B of ' Aa, A Ba. (A /\ B)a, n n ( A A.)a = l l Al A◊:l n n t:, A. )a l l ( = t:, l A◊: l We can further generalize the application of the definition to get the iw~ge under a. of a function of a set of el ement~ of ii:..,, consisting of successive joins ana. meets of these elements, eq_ual to the same function of the respective i mages of the elements. The ic3.entity automorphism l, ~ A Thus A<. l, for every A €: ~ If (11.a) ~ a = and Aex,~ ~ A is given by: = A. are t wo automorphisms of L . By the product O'.~ , we write we mean the result of app lying 4 ex,, fo llowed by S acting on the elements of aS A :By the inverse -l CX, -l A a, CX, for a ll . Aa.S ,,, -< of CX, (; ~ Evidently A E~ £ , and ,..,e write: we mean: A -l a, a, = -l -· (JJ1, L . We close this section with the fundamental THEOREM 3.1. -;/!.,. The set .i, of all the automorphi sms CX, of a lattice form a group, the automorphism group of the lattice. The theorem is easily verified and the proof is omit ted. 4. nmd.amental properties of automo rphi sms. In thi s section we shall state a number of lemmas most of wh ich are obvious from the definition of an automorphism of a lattice. LEM~IA. 4.1. An automorphism LEMMA 4.2. Chains are preserved by The element s a, preserves t he relations ) and ( M and J and the nodes a, of N.1. of £.. , ;J:.,,.. LEMViA. 4.1-~. The Dedekind. propert y is preserved. by every LEMMA ~- .i::;. The a.istribut ive property i s preserved by every LEMMA 4.6. Complement ed elements are carrie,l in~o compleme nted. elements by every LE}~i.A. 4.7 . :;:,, and • a, • a, • a, . .An automor-9hism a, preserves the covering rela tions < . LEMMA 4.8. a, . a, . when they exist, are inva riant under every b;y We omit the proofs . Descend.ing a nd ascending cha in cono_i tions are preserved. 5 LEMM.A. 4.9. If an element is expressible as the join of points, so too is its image under a,. We note that if A contains only the contains only the n X1 , X2 , points n points Let ~ X n a,. In a finite lattice with rank function defined, rank is preserved by Notation. • T'nis important fact is clear from the 1-1 ordering imposed by LEMMA 4.10. • a, • For the following lemma we adopt a new notation. be the number of elements covering the element °-x be the number of elements covered by X • Then, X and and °-x denote the number of bonds emanating upward and downward, respectively from X in the lattice diagram. call X a (5 ~ If = 5 and °-x = 7, we shall 7) element. M is then an (n - 0) element, where n is the number of points; J is a (0, m) element. We shall use this notation again later on. LEMMA. 4.11. An (r - s) element is carried into an element by an automorphism of (r - s) ,;l . We conclude t}).is section with the proof of the simple but important THEOREM 4.1. An automorphism a, carries sublattices into sub- lattices of like kind; i.e., distributive sublattices go into distributive sublattices, point sublattices are carried into point "' sublattices, and so on. Let elements of uo be a sublattice of -J:, , and a, uo be the set of L corresponding under a, to the elements of Uo . 6 If A :s and = are any elements of U0 . Hence €: '1,.(. ~ • Similarly U is a sublattice of CL 0 closed with respect to join and meet. Th8,t 'u,0 , since it is .,, J a., and. '-1., are 0 sublattices of like kind follo ws i mmedi R-te ly from the precedine; ler:imas. 5. lattice Permutation uro-oerties. £.. The automorphism group is a permutation group on its elements. is the image of A under some conjugate elements and write a, of A rv B .1J of a :S = Aa., If /J , we call A and B The relation of conjugacy is V\ clearly an equivalence relation. It enables us to divide ,t_ into classes of conjugate elements where the set 111.i consists of all im1",ge s of some Ai E ~ . These classes a :ce then the transitive systems of ,,b vm.en it is represented as a permutation group on the elements of L . Con si der a class ' An • 1n_ of eq_ui valent ( conj ugate) elements From well kno,vn theorems* of group theory we get the next four theorems. THEOREM 5.1. are a ll elements of of (A 1 A3 ), , then the symmetric group J. are a ll elements of group of JJ , then t~e a lternating group ! n • • ·, (A 1 A ) n is a subgroup is a sul)- ~ THEOREM 5. 3. T~rn set of all automor-_phi sms leaving an element .A 1 11 Introcluction to the Theory of Groups of Finite .Order, 11 Carmichael Corollary to Theorem II, p. 8; Corollary to Theorer.i IV, p. 11; 7 unaltered forms a subgroup of .iJ n A1 • the number of images of h has pri:ne orde If element l, of index p COROLLARY 5.3. The index of this subgroup is p , this subgroup must be the identi t;? or ,k itself of index one. If the automorphism gro1.,p Hence we have b of ~ has pri me p , ever-J element is either invariant or has exactly order p iimages, including itself. ..iJ is a prime p , the c l asses Hence, when the order of of con juga te elements of £ each . consist of p element s or one element The e l ements of ea.ch class of order an automorphism a, Either p may be so ordered that will be the same permutation on each class. wi ll permute a ll element s of ea ch class or a, a =/: ~ if of a cla ss. Otherwise we could write permutat iom on th e A. Then the order o: a, a, = L , it i s a cyclic permutation on all the Furthermore a, A.1. as a product of cycli c no t wo of which have a letter in co:m.mon.* 1. would. ·be the least common multiple of the degrees of the cycl i c permut ations which com:9ose it.** since the order of 1r{i a is This is a contradiction, p , a prime . The remaining elements of J!t ca n be written as powers of .a,, since a group of ::;iri:ne order is cyc li c ancl is generated by a ny e l ement other than the i dent it y. where k Hence we can wri te has one of the val ues 1 , 2, Now consider the automorphism groc:p J.t ~ = (:p - 1 ). of a point lattice ;/:, • Since covering is preserv ed, the points are perrmt ed amone; themselves 7. Ibid. Theorem I, p. Ibid. Theorem V, p. 11. 8 and may be divided into classes of conjugate elements. Tha t ever_x =/=- l, will permute po ints is clear from the defi nition of a :p,aint a., lattice and the f act that cha ins a re p reserve d.. THEOREM 5.4. Hence we have The automorphism. group of a point l att i ce is a permutation group on its points. COROLLA.RY 5.4. The automoryhism group of a point lattice of 1 n Jloint s is a subgroup of the symmetric group- 6. I nvariant sublattices a nd subgrou:ps. of Birkhoff. * form a sublattice. The set of el ements of co r responcts a sub lattice of a., cf. invariant under J.I. of the automorphism group .J/ 1A_. consisting of a ll elements invariant 1/--the centra li zation of •• )){. = n l , ano. if U ~. If , 1f l ' 0 J.f0 , respectively, then C Un = £. a sublattice of Jf1 , Ui + l (i = 0, 1, 2, • • k = . . . lfn a re t he centralizations corre ~ponding to 11.,i a., This sublattice is ca lled the centralization of More general ly, to a subgroup under every We state here a theorem Its proo f is left to the reader. THEOREM 6.1. a., • .4 n • ' Z{,_[l with (n - 1)) . The a na lo gue of this theorem is the following theorem for which we shall have use l a t er . TIDJOREM 6.::>. gi ven element Tue set of a ll automorphisms of A una lte red forms a subgrou-p of to a sublattice ./.J leavi ng a .!J . More generally, U of £, correspond a subgr oup JI.- of ~ v,h ich l eave s it absolutely unaltered. Birkhoff (3) -- 11 011 the Structure of Ab stract Al gebra s, 11 p . 435. 9 The first part of the theorem is merely a restatement of Theorem We note tha t the subgroup Jf t ha t leave s a su·bla ttice absolutely lLQaltered is a subgroup of a l a rger subgroup ;¥,0 wh ich U u.11.e,lterea. merely as a sub l a tti ce. lea ves TIDlOBEM 6.3. with Zl. A :J B Let A B and be t wo invariant elements of Then the q_uot ient lattice A B £. , is invariant a s a £. . sublattice of This theorem is an i mmediate conseq_uence of Lemma 4.2. COROLLA.RY 6. 3. • ., A n Let of ?'J1. be e, cl ass of conjugate elements L A A. h l !::,. invaria nt as a sub l att ice of n n .tor Ll. A . and !::,. A . i = Then i l al . is ·\ l are invariant under every automorphi sm. l We need not restrict Theorem 6. 3 and. its coro llary to the who l e automorphism group ./t . We can consider q_uot i ent latti ces invariant u.11.der a subgroup ft of is the following: If A ) B ".nd o A more general re sult than Theorem 6.3 m1der rt, , with Aa .,, ..., T:,a .o , t'nen Definition. ~. .!J . Le t Ni, N2 , • , Nr J ':) N :, N r r - 1 :> N r J Nz lattices ,-) ~ ) • ) N Nr M Ni r - 1 sublatt ices (Ha up tu:at erve rbande) of with .. be the nodes of a lattice ::, Ni :) M . The quo tient a re ca lled the nrinciual cl .n·om Lemma 4.3 and Theorem 6.3 THEORZ'v! 6.4. re ge t i mmeclia tely The principa l sublattices of as sublattices under the aut omorphisms of L, are invari ant 10 The aut omo rphisms of any one pri ncipal subl a ttice a re certainly independent of those of the other principa l subl a ttices. We therefore conclude 'l illlOREM 6. 5. The auto mo rphi sm grou;_o 1 1J of a l a ttice ;f_ with one or mor e nodes i s the di rect product of the groups of its prin cipa l subla ttices . .!from t his it is seen t ha t i n the study of l a ttice a utomorph ism groups we can conf ine ourselves to h .ttices without nodes. We shall make use of this -impo rtant theorem l a t er on . 7. Point l at tice with the symme tric group. In this section i s g iven a ne ce ssary a nd. sufficient condit ion that a :point l attice with n points have the symmetric automorphism group Definition. Le t J n . £, b e a n Archimaed_ian lattice in which all chains between an arbi tra.ry :pair of depend.ent elements are of eq_u.al length . We shall call £ an equal -chain l atti ce (a.usgeglichener Verba.no. ). Notation. We shall denote the points of a po int lattice by numbern 1, 2, • • •, n. Since each element i s the join of a ll the points it contains , we shal l incorpore,te thi s i d.ea in our not a tion . If an element conta.ins 1, 2, wr i t e 12~-7 i nstead of 4 and 7 1 1\ 2 /'\ l+ /\ 7 . a po int l attice tee j oin of 12479 and a nd no other points, we shall Then by the properties of 2589 is 1245789 . We use this notation in with n THEOREM 7.1. Let J_ be an eq_ual chain po i nt l attice of r ank n Let each rank consist of a.11 the combinat ion s of the p o::..nc s . • .j. points taken k at a ti!cle, k f obvio'l."csly inc1·er~sing with the ro,nk r . 11 Then, as < r takes in the suc ces s ive values P - 1). e ( p - 1), n uoints c: k In o ther words M is of rank zero, the 1, 2, are of r a p.k those of re.nk •' n 12, 13, • • • f- 1 If ana_ rank ! p - 2)' t,-,.kes on t he successive va lue s , l, 2, J 1 p- 1 a t a time; then k a lone of rank f 2 consisting of the juinps to the value n for ~ n _:Points r = p . skipped. an integr2.l vah.e except between the rank k a re the , (n - 1) n ; and so on 1 n, 23, • till ,,,e have the elenent R of rcmk and we get 2)' r esp ectively. elements taken l, 2, we would not have m1iq_ue join of elements. p- 1 This is best illustra.ted by the example of Fi 6ure 1. M Figure 1 . Th5.s is not the cliagram of a lattice because the join, se.y, of 2 is !10 t 1.miq_ue. 1 and. 12 THEOREM 7.2. A ~o oint l a ttice , symmetric group £ with n po ints has the as its automorphism .o:ro ,·_p if and. onlv if it "' .; .,<) ~1 is an eq_ual-chain point l attice with each rank consisting of all the C~ combinations of the n points taken for each re.nk, hence if ardonly if Theorem £ k at a time, k constsnt is of tl:e form given in 7.1. Supp o se Jn is the group of automorphisms of £ po i nts cont a ined in an arbitrary element Then in our notation A Since p oints, v e see that by te..king the n Aex, points A be 1, 2, • • • r . = (123 • • ·r )ex, = 123 • • • r . ex, Let the ranges ove r all pe rmutations of the n ranges over all the en r r el.. consists c f a number of such sets with different at a ti me . r . Thus elements represented It is clear that each set is of con::.ta.nt rank so that ~ i s a n eq_ual -chain point lattice . The remaino.er of t he :proof of the necessity follows i mmecliate l y from Theorem 7.1. The sufficienc~r is evia_ent from the symr.1e tr·ical way in which the elements are represented in terms of the When point s . point s . e = n in this type of lattice we have the :Boolean algebra consisting of all subsets of (1)). n n points. (See Theorem 24.2 , Birkhoff This l att ice is the distributive lattice generated. by The c.iagram of the Boolean a lgebra with in the fi gure on the next page . n = 5 n i s given 13 M Fig,11.re 2 When p = n - k, the lattice of Theorems 7.1 and 7.2 is obtained from the Boolean algebra of elements of rank for which n points by omitting the n - k, n - k + 1, • • • , n - 1. f = 3 and n = 5 is illustrated below. M Figure 3 A simple case 14 This type lattice is seen to be a Birkh0ff lattice in which the ~ cond.ition holds.* The condition fails for the elements covered by f' = 2 except for the ce.se J In the latter case the lattice is modula r, but is distributive only when n = ,.., c:. • A cl.ie,gram of this c;:i._se is given in Figure l.~. • I Fi gure 4 The Boolean algebra is the most interesting c~se of this type of lattice. Every distributive lattice of rank of the Boolean algebra of THEOREM 7.3. rank n n points. n is a sublsttice From this follows The automorphism group of a distributive lattice of is a subgroup of the symmetric group ,J n . In the next three sections we shall be concerned with the automorphism groups of distribution lc. ttices. * ]irkhoff (1)-- 11 Cn the Combinc.tion s of Suba l gebras, 11 p. l.i.45. 15 8. Notation for distributive lattices. The notation we now develop for the elements of a distr ibutive l att ice useful in studying its automorphism group . e will be We m,e Theorems 17.2 and 17.3 of BirF~_off 1 s paper.* The genen).tors are 12.beled in order of r ap_k . The n point generators are repr esented by t he integers l, 2, 3, • · • • These of course, generc,te the Boolean a l geb r a I n e of r a nk n of n Sect i on 7. ½i Then, "by Theorem 17. 2 ment ioned 2.bove each el ement of is identified with a subset cf the that the join of two elements X and. l ogi ca l swu of the subsets of X common part . 3 ) , ( n points i n such a way Y i s identified with the and. Y anc, the meet with their Now consider the generators of r ank suppose there are (n - n - p of them . + antl If they cover, say, 1, 4 , 7, • • • 1) respective l y we denote them by (n - 3)(n 7( n + 3), • 2 l( n + 1) , 4 (n + p - 1) , and (n - l)(n + p ) 2 ) , respectively. The element s generatecl by these new elements 2.na_ those of are /f \..,n t:otten by taking the logical sum as the j oin and the common part c:ts the meet . We obtain the representation of the generators of third. and_ higher orders in the same manner, 2nd ge t the r esulting generated elements as before by taking l ogic8.l sum anfl~ comr:ion part . for join and meet re s,e ct i vel y. 135789 13456789 Thus if' the . ,_ we designate l t, bv ., ancl (1 35789 ,1468) rth 135789.tt . = 18 . generator covers Also 135739 I\ 1468 = It must be re:oembered that the number of the gener~tors is e~ual to the r ank of the distributive lattice . Birkhoff (1) --"0n the Combination of Subalgeoras, 11 pp . 451.i--456. 16 To illustrate this notation the diagram of a lattice of rank 10 is given below. Its generators, enclosed by squares, are 1, 2, 3 of rank 1; 124, 125 covering 12 and 236, 237 covering 23 all of rank 3; 12345678 of rank 8; and 123456789, 12345678tl of rank 9. M Figure 5 17 The group of this l attice is the direct product of the four groups {, . (9t)} , {L, (12)(47)(56)} , therefore of orcler 16. {L , (45) J , and {L, (67)} . It is It is noted that the four generators 124, 125, 236, 237 form a sinrply transitive system with the two sets 124, 125 and 236, 237 as systems of imprimitivity. This notation h~s several inrportant useful properties. The rank of an element is merely the number of digits or letters in its representation. Furthermore, we can tell at a glance which genera.tors are contained in a particular element. Now the structure of a distributive lattice is determined uniquely An by the manner in which transitivity is ascribed to the generators. automorphism will then permute certain generator s of the same rank, since the genera.tors determine absolutely the coinIJlete structure of the lattice. Using the notation here developed, an automorphism a, will then be a permutation on the end digits of the representations of the generat ors . .Applying this permutation to the represent a tion of a.n element automatically the image Ao, 9. .A we get of A. Direct product ,of linear lattices. linear lattices of rank m and The direct procl.uct of t wo n respectively, is represented by a rectangular checkerboard diagram. We show the diagram of the direct product of the linear lattices of rank 3 and 5 in Figure 6. 13 .E'igu.re 6 The direct product of n linear lattices of rank f1, Pn respectively, is represented by the projection upon two dime~sions of a generalized rectangular network structure of this type is a g.c.d.-1.c.m. positive integer n where n dimensions. A lattice of The elements are factors of the lattice. f2 ri = n P1 P2 . . . Pn Pn The join of two elements is ta.ken to be the l.c.m. cf the elements, while The generators are Pl, Pl 2 , their meet is their g.c.a. p e1 P2 '2, e2 , • . . . .. P P J 1 n' 2 . • • • , Pn n ' The automorphism group of this lattice is easily seen to be the direct product of symmetric groups: X • • • X where is the number of (i) r.J. (ii) the Jr. l ( iii) r1 = 2r 2 + 3r3 +• • pj equal to i are deleted when r. = • + kr = p = f1 + k J. 1 p2 + • • • + e._n A special case of this type of lattice is the :Boolean Algebra of rank n of Section 7. This lattice is the direct product of en· n linear 19 lattices of rank 1. }J Hence r1 = n and O = r2 = r 3 = • • • so that ..J n , as we ha.ve alre~:;,dy seen. = Suppose we have given the direct product e of k such Boolean e. of rank n1, n2, . . . ~ respectively. Since each e. is the a_irect product of ni linear lattices of rank 1, e, is the direct algebras 1 1 • + ~ linear lattices of rank 1 and therefore product of k ~ is the Boolean algebra of rank ni J~n.. with group :_ I I 1 This example illustrates the fact that the direct product of the groups of several lattices is merely a subgroup of the group of the direct product of these lattices. The simplest case is given below. J I I [!,3] [2,l] &,ril) A r1 }J = ( L ,(n.)) h = (L} [M>M) J., -:; ,,13 Figure 7 10. Lattice with given group. In the preceding sections we have dealt with a few of the aspects of the problem of finding the group @fa given lattice. Finding fille group of a lattice is a separate problem for each lattice in the most general case. We can establish significant 20 theorems on the group of a lattice only for well-behaved types such as equal-chain lattices and distributive lattices. In this section we shall be concerned with the converse problem, the construction of a lattice naving a given group for its automorphism group. f i,-,it• The problem has been solved for the case of"abelian groups. We state this result now in li11ite. THEOREM 10.l. J . There exists Given an arbitrary"abelian group £ a distributive lattice }J with as its automorphism group. f it1,t.e Any abelian group is the direct product of cyclic groups.* /\ Theorem 6.5 states that the group of a lattice with nodes i s the direct product of the groups of the principal sublattices. Our problem is therefore reduced to constructing the principal sublat t ices having the cyclic groups for automorphism groups. We do this using the notation developed for distributive lattices in Section 8. Case Let n = 2. n be the order of the cyclic group. We take ~the Boolean Al gebra of rank We omit the cases n = 3 Case n = 5. and n = 4 for the time being. We take as generators the set of five points, 1, 2, 3, 4, 5, the set of rank three 12a 3 , 23a of rank seven 1234a 3a b 4 Let a (See Figure 7). 2. , 2345a a b 4 5 5 , 1 , 4 34a 3451a a b 5 45a , 5 1 , 2 , 1 5la , 2 and the set 4512a a b 3 , 5123a a3b 1 2 2 be the permutation (12345)(a 1 a 2 a 3a 4 a 5 )(b 1 b 2 b 3b 4 b 6 ). • 4 On acc ount of the cyclic manner in which we have defined the generators, a., and its powers ak (k = 2, 3, 4. 5 with a 5 = i lattice which we shall denote by permuted cyclically by ) are aut omorphisms of the distributive (! 5 • Each set of generat ors is a.,. Carmichael-- 11 Introduction to the Theory of Groups of l!.,inite Order, 11 p. 66, Theorems XIV and XV. 21 We must now show that there are no other automorphisms of The five powers of 2 beginning with the first carry the points a into the ordered pairs 2,3; 3,4; 4,5; 5,1; 1,2 respectively. 1 and We now show from consideration of the second and third sets of genera.tors that only these powers of a ce.n do this. Since the three sets of generators must be permuted among themselves, it is evident from the second set that consecutive integers must be carried into consecutive integers. If ~ exists carrying 1 and 2 into 3 and 2, 4 and 3, or 5 and 4 respectively, 20-l 3 -l a$ -1 , a~ , or a~ then will transpose 1 and 2. In any event we need consider only the existence of an automorphism transposing 1 and 2 . If one exists 12a 3 is invariant, 23a 4 ~ 15a2 so that get the permutation images of the b. 5 We find ---+- 4512a 2 a 3 b u • But there is no such genera.tor for that 1234a. 3 a 4 b 5 any value of ~ To determine the we consider the third set of generators. 1 3 u. Hence there are no other automorphisms than those of a the cyclic group For the case , and we have completed. the case n ::> n = 5. 5 we apply the same argument to the analogous sets of generators 1, 2, • • • , (n - 2)(n - l)a n This me thod of choosing generators breaks do,~1 for the case n = 4. '7 To get a distributive lattice ~4 with the cyclic group of order 4 as automorphism group we take for generators the following four sets: 1, 2, 3, 4; 12a 3 , 23a 4 , 34a 1 , 4la 2 ; 12b 3 , 23b 4 , 34b 1 , 4lb 2 ; and 123a b c 3 4 , 1 234a b c 4 1 , 2 34la b c 1 2 , 3 412a b c 2 3 . 4 The four powers of 22 Furthermore, from the second set of generators we see tha t a pair of cyclically consecutive members of 1, 2, 3, 4 (as elements or subscripts of elements) must go into a pair of consecutive numbers in the same or inverse order. If a pair goes into a pair in the same order, it is easily seen that the resulting automorphisip. must be a power of a,. If the pair goes into a pair in reverse era.er, we need. consider, as we did in case n = 5, only the possibility of a transposition of 1 and 2. If one exists we get the permutation upon consideration of the second and third sets of generators. 123a 3 b 4 c 1 ~ 412a 3 b2 c u Hence the powers of a, :But this means that which does not exist for any value of u . are the only automol1)hisms of this lattice. We must take even more complicated (comparatively) sets of generators for a distributive lattice with automoriJhism group cyclic of order three. are automorphisms because of the cyclic order in whi ch the l ette rs appear in the generators. into 4, or if If ~ exists carrying 1 into 5 then ~a,2 carries 1 P exists carrying l into 6, ~a carries 1 into 4. automorphism exists carrying l into 4, then 124a 3 a3 is invariant. We have two subcases: 4 ~ l ~ 124a 3 and 4 ~ 2 If an and 4 --J- 2 . The latter is impossible, since 1 --t 4. If 235a 1 ~ 316a 2 , and 316a2 ~ 2358-1 • Again the latter is impossible since 1 --t 4 . then 2 ---, 1, We have excluded the possibility of an automorphi,sm carrying 1 into 4, 5 or 6. 23 Similarly 2 and 3 dannot go into 4, 5 or 6. Hence 1, 2, 3 are permuted among themselves and 4, 5, 6 among themselves. If 1 is inva riant and 2 and j are transposed, then 235a 1 and 5 are invariant and 4 and 6 are transposed from a consideration of the fourth set of generators. But in the fifth set 1257b 3 ~ u. 1357bu which does not Hence no automorphism leaves 1 invariant while transposing 2 and 3. Similarly no automorphism transposes 3 and exist for any value of 1 or 1 and 2. On t.he other hand, the automorphisms that permute 1, 2, 3 cyclically are easily seen to be the three powers of a. =~ a, a 2 , a 3 Therefore, constitute the cyclic automorphism group of order three of this lattice. We have now completed the demonstration of the existence of distributive lattices with cyclic automorphism groups of all orders. Given an f inite arbitrary" abelian group , we express it as the direct product of cyclic groups. We then construct a distributive lattice with nodes having for its chief sublattices lattices with these cyclic groups as automorphism groups. This distrioutive lattice is the one required by the theorem and the proof is complete. It should be emphasized that this method of construction is by no means unique. We could have constructed lattices with cyclic groups of order 4 or greater in the same manner as we constructed the one with cyclic group of order 3. We chose a simpler set of generators for order 4 and a still simpler set for order 5 or greater. The theorem is merely an existence tneorem. Tne construction of lattices having various types of non-abelian automorphism groups will be the subject of further investigation by the author. 24 11. Two examples. The diagram of Figure 8 is tha t of the so-ca lled "symmetrica l equivalence lattice o! ct.egree four )l point (a nd dual point) lattice. * . This is a modula r Ih: f a ct Its group has order 24. its group must ue isomorphic with the symmetric group J 4 .** The elements are listed in Table I. Figure 8 a,: (14) (jti) a: (25) (1364) 1: (124)(356) 2 ra: (13)(46) -a: ( 25)(1463) 1: (124) 2 (356) Y: ( 24)(35) b: (16) (2354) 2: (1.3::,) (246) 2 a,: (23)(45) b: (16)(2453) 2: (135) 2 (246) B: (25) (34) ra : ( 15)(26) c: (34 )(1265) 3: (145) 2 (236) C: (16)(25) -y: (12)(56) -c: (34)(1562) 3: (14~ )(236) 2 (,: () A: (16) (34) -'- 4: (123) 2 (456) 4: (123) (456)1Table * I Birk:hoff (3)-- 11 0n the Structure of Abstract Algebras, 11 pp. 436- 7 and 11'18 , pp. 447-8. *"' Ibid. 'l'he o r em 2j, p. 449. 25 The second example, given in Figure 9, i s that of the smallest non-trivial projective geometry. and seven d.ual points. It is of rank 3, with seven point s It is surprising that its automorphism group turns out to be the smallest non-cyclic, non-alternating simple group. Projective geometries and simple non-cyclic, non-alterna ting groups a re few in number. Whether or not there is a relation between them as hinted by the example will be a subject of later investigation. The automorphism group, as a pe rmutation group on the points, is doubly transitive. It is generated by (1423576) and (23)(56). the elements in Table II. l'2. 3 31,7 I 7 M Figure 9 We list 26 1 ---, 1 23 - t 23 23 --t l2 23 _,. 7 23 -+ 74 23 -I' 56 23 --t 65 (56)(47), (57)(46), (45)(67) " ' (23)(56), (23)(4675), (23)(4576) (23)(47), (24)(37), (2437)(56), (245)(376), (246)(375) (27) (34), (2734)(56), (275)(346), (276)(345) (25)(36), (2536)(47), (254)(367), (257)(364) (2635)(47), (26)(35), (267)(354), (264)(357) 1 ---+ 2 23 --t 13 23 --t 31 23 - t 46 23 ~ 64 23 --I) 57 23 ~ 75 (12) (45), (12)(4756), (12)(67), (12) (4657) (12a)(467). (123)(576). (123)(475). (123)(456) (12 3657), (124)(367), (1245)(36), (1247536) (1263475), (126)(345), (1267)(34),. (1265734) • (12537~6), (125)(376), (1254)(37), (1256437) (127)(354), (1273564), (1276)(35), (1274635) 1 ----, 3 23 --> 12 23 --t 21 23 ~ 23 --t 23 -t 67 23 ~ 76 (132)(465), (132)(476), (132)(567), (132)(457) (13) (4567). (13)(a1>. (13) (lJ.76a). (13) (46) (135)(246). c1 52 76). (135672 ). (1357)(24) (1342567). (13 )(257). (134762a>· (1346)(25) (1375) ( 26), (1374526), (137265 ) , (137)( 264) (1365427), (1364) (27), (1362745), (136) (275} 1 ---t 4 23 -t 26 23 ~ 62 23 ~ 35 23 ➔ 53 23 - t 17 23 - t 71 (147)(365), (14)(36), (145)(367), (1436)(57) (1432675), (1457326), (14)(2653), (147)(263) (1467235), (1423576), (147)(235), {14)(2356) (14) (25), (147)(256), (1425)(67), (146)(257) (1452) (37), (1437562), (1465372) , (142) (376) (1427653), (1463)(27), (143)(275)~ (1456273) 1 ~ 5 23 ........ 16 23 ..-,. 6i 23 -+ 27 23 -+ 72 23 --f 34 23 ➔ 43 (152)(367), (1574362) , (1536472), (1542) (36) (153)(264), (1547263), (1526743), (1573)(26) (1537)(46), (154)(376). (15)(37), {156)(374) (15)(2743), (156)(273), (1532764), (1546327) (15)(2347), (156)(234), (1523467), (1576234) (1524)(67), (157)(246), (15)(24), {156)(247) 1 --, 6 23 _. 15 23 ~ 51 23 -, 24 23 ~ 42 23 ➔ 37 23 -t 73 (1647352), (162) (354), (1672) (35), (16;3571+~) (1674253). (163)(257). (1643)(2 >. (1625473) (167)(345), (1634)(57), (165)(3 7), (16)(34) (165) (243), (16) (2473), (1675324), (1632457) (1645237), (1623754), (165)(237), {16)(237~) (165) (274), (16) (27), (164) (275), {1627) (45) 23 -, 14 23 --+ 41 23 ~ 25 23 --t 52 23 -+ 36 23 ---t 63 (1734652), (1-762)(34), (172)(345), (1756342) (1753)(24), (172ij563), (1765243), {173)(246) (174)(356), (17)(35), (1735)(46), (176)(354) (1764325), (1732546), (17)(2563), (174)(253) (1754236), (17236ly5), (17)(2365), {174)(236) (174)(265), (17)(26), (1726)(45), (175)(264) 64 1 ---t 7 = ' 4 4 Table II III Supplementary Theorems 12. Modular lattices.. In this section will be given some supplementary results not directly connected with automor-_phisms of lattices. first a proof of a theorem of Birkhoff.* We give His proof appears incomplete The proof given here is along a different line. If X1 , X2 , THEOREM 12.1. • • Xw are the points of a finite , w y :_( A 03 , and if modular lattice • X. , then Y is the join of the X. l l. l. which it contains. To prove this theorem we make an induction on the rank. is true for elements of rank r = 1. rank r = n. Let Al X.l. in and let covered by Y. 1, 2, Yi, Y2, Then, since x~tr ( rr = 1, 2, Assume the theorem is true for Y be an arbitrary element of rank w . . . ' m). y = r = n + 1 contained . . . ym be the elements (of rank r = n) y Y,r C . . •) f'. <r ) The theorem ,,., w <: ~ 3E. y(f" = A ' ,r= 1 are the points contained in Y,r l l. m Al Y.l. the join of the Xtr,r = X(T,r where ( <T = ' contained in the Yi. If X0 is any other point not included among the x,, m }!otr XI\L\, = X0 / \ y has rank n + 2 by the law of rank Xinr A 0 <T= 1 tr= 1 y , y contains only the points X <TIT in a modular lattice. Hence X 0 ¢ and is the join of these points. rank r Since y was an arbitrary element of n + 1, the theorem holds for all elements of that rank = w contained in Al X.l. The induction on r ( r = 1, 2, . . . , w) is now complete. COROLLARY 12.1 A. The QUOtient sublattice w lattice. If L\. l X. J then d!, is a point is a point lat t ice (and also a l. dual point lattice). Birkhoff (1)-- "0n the Combination of Subalgebras, 11 Theorem 11.).1-, p. 449. 28 The statement of the parentheses is gotten merely by dualizing the proof of the theorem. COROLLARY 12.1 ]. If F1 , F2 , • • •, Fw are the elements covering F in a finite modular lattice (B, and if Y is an element an element w F.J. which it contains. y F.J. :, L\ l satisfying the relation :> y then F 1t The quotient lattice F.J. F is the join of the is a point lattice and is the dual of a point lattice. In the following theorem we employ the notation introduced for Lemma 4.11. THEOREM 12.2. In a modular point (dual point) lattice (1 - k) elements are the du.al points (elements covered by the only (k - 1) elements are the points of Let where Xi (8. :By the law of rank the elements Y. = Y /\ X. , They must then all be equal to an element, say J J. J. ranges over all points not contained in · Y, all cover all the points it is d3 ; J ) of Y is the join of the points which it Y be a (1 - k) element. contains by Theorem 12.1. (1.3 the only so that that there can then be only one Z. Y is a dual point. Since Y. Z contains We note in ~assing X.J. not contained in y . The second part of the theorem is merely the dual of the first part, since a modular point lattice is its own du.al. COROLLARY 12.2. A modul ar point (dual point) lattice has (1 - 1) elements if and only if it is of rank two. The following theorem is a useful test for the modularity of a lattice when the lattice diagram is given. THEOREM 12.3. lattice CB . Let Let We employ the notation of Lemma 4.11. Y be a (u - d) element in a finite modular • • • X d be the elements covered by Y 29 2.nd Yi, Y2 , d Dy =L • • • , Y ('1xd' - 1) = O'"= 1 be the elements covering u Y. Then u 2. ,,r= 1 (dy - 1) :: Uy • /f" This theorem stat es that in the lattice diagram the sum of all the bonds emanating downward from the elements covering Y diminished by the ntunber of these elements is equal to the number of bonds emanating upward from the elements covered by Let xi (I' Y diminished by the number of these elements. X tr'1x = Y ' x2tr Y, (J" '1x"" elements covering X6 • be the Y., Figure 10 icr These X tT are all distinct for every o and i #: 1 . woulcl be no unique join of t wo X f1 covered by a common Otherwise there X~ ( "'F Y) indicated in Figure 11. Now in a modular lattice when A and cover (A, B), then A I'"\ B covers both Y -- x'I( -x' ~ Xot Figure 11 as B each 30 itr =I: Hence the elements X I' f\ Y (id" A and ] . are among the cover 1) These (1½(.,. - 1) elements '1ccr ) then account for all the and Y and = 3t 2, • • (1½(0- - 1) bonds emanating upward. from the - 1) bonds emanating downward from the in each case bonds emanating from Y Summing on Ylf" excluding we have the (f" inequality u 2. - 1) . tr= 1 Dualizing the argument we get Uy < Dy • d Dy Hence u - <T=2. l ('½ca- - 1) = z:. rr= 1 (~rr - 1) :::: Uy and the theorem is proved. We note that than Dy = Uy represents the number of elements other Y which are covered by elements covering are elements other than 13. Conclusion. Y and that these elements Y covering the elements covered by Y. Several of the results of this paper will be further developed in a later investigation. The problem of determining the group of a distributive lattice will be carried further. Also the problem of constructing a lattice, given its automorphism group), will be extended to certain non-abelian gro-o.ps. Another problem will be the investigation of the relation of normal divisors of the automor:ohism group to the structure of the lattice and rela ted ideas. The further study of the automorphisms of lattices will he confined to distributive and other specia l types of lattices where results are more likely to be significant. • 31 REFEREi'JCES A. A. Albert 1. Modern Higher Algebra, The University of Chicago Press, Chicago, 1937. G. Birkhoff 1. On the combination of subalgebras, Cambridge Phil. Proc., vol. 29 (1933), pp. 441-464. 2. Anulica tions of lattice algebra, Cambridge Phil. Proc., vol. 30 (1934), pp. 115-122. 3. On the structure of abstract algebras, vol. Cambridge Phil. Proc., 31, (1935), pp. 433-454. R. D. Carmichael 1. Introduction.!£ the Theory of Grou~s of Finite Order, Boston 1937. R. Dedekind 1. Uber die von drei Modula erzeugte Dualgru.ppe, Ges. Werke II (1931), Abh. XXX pp. 236-271. F. Klein-Barmen) 1. Beitrage zu.r Theorie der Verbande, pp. Math. Zeitschr. 39 (1934), 227-239. 2. Gru.ndzuge der Theorie der Verbande, Math. Annalen 111 (1935), pp. 596-621. 3. Uber ausgeglichene Verbande, Math. Annalen 112 (1936), pp . 411-18. 4. Dedekindsche und distributive Verbande , Math, Zeitschr. pp. 41 (1936), 261-280. 5. Birkhoffsche und harmonische Verbande, Math. Zeitschr. 42 (1936), 32 A. Kurosch 1. Durchschnitts Darstellungen mit irred.uziitleKomponenten in Ringen ~ vol. in so gennanten Dualgruppen, Trans. Moscow Math. Soc., 42 (1935), pp. 613-616. E. Netto 1. Theory of Sub sti tut ions, ( translated by F. n. Cole), The Inland Press, Ann Arbor, Michigan. 0 . Ore 1. On the Foundation of abstract algebra.!_, Annals of Math., vol. 36 (1935), pp. 406-437. B. L. va n der Waerden 1. Moderne Algebra l, Berlin 1930. H. Weber 1. Lehrbuch der Algebra, vol. II, Braunschweig ,18.99. •