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CONDITIONS UNDER WHICH A LATT ICE IS ISOMORPHIC

TO A LATTICE OF SUBGROUPS OF A GROUP

B. V. Yakovlev UDC 519.51:519.45

Suzuki [ i] has posed the prob lem of f inding suff ic ient condit ions underwhich a la t t ice is isomorphic to a la t t ice of subgroups of a group $. Anishch -

enko [3] has obta ined such condit ions (which are a lso necessary) in the case

that $ is an Abeli an p-gr oup with not more than two generatr ices. In this paper

we formula te in terms of la t t ice theory the nece ssa ry and suff ic ien t condit ions

under which a la t t ice is isomorph ic to a la t t ice of subgroups of a group (3,

Theo rem 6). They follow from the condit ions un der whic h a la t t ice is isomorph ic

to a la t t ice of subgroups of a f ree group of rank not smaller than two (3,

Theor em 5), and f rom a la t t ice - theory chara cter i za tio n of a normal divisor of a

free group (3, Theor em 4) . As an applic a tion of the obta i ned results we shall

prove that any la t t ice iso morph ism of a nonco mmuta tive group in whic h any two

elements genera te a f ree group can be induced only by one group isom orphi sm (4,

Theor em 7). Hence, follows Sadovskii 's theorem [4] on the str ic t def init ion of

a nonc ommut ative locally f ree group by a la t t ice of subgroups.

Everyw here in the following pape r we shall denote by L a complete la t t ice ,

by o the null elem ent of L, and by L (G) a lattice of subgroup s of G. The order,

sum, and produ ct of two la t t ice e lements o and ~ wil l be denoted by the symbols

~ , a+ ~ , wi th the symbols ~ ,u ,n be ing used in a se t - theore t ica l sense . I f

M ~L, then ~ Mw il l denote the sum of a l l the e lement s belong ing to ~ .

Followi ng [5] , we shall say that an e lement ae L is cyclic if the interval

~//0 is a distr ib utive la t t ice with a maxi mali ty condit ion. Such a def init i on

is related to the fact that if HeL(O), then H will be a cyclic e lement of L~)

if and only if H is a cyclic subgrou p of G. This easil y foll ows from the re-

sults of [6]. The se t of a l l cyclic e lements of the la t t ice L will be denoted

by [~), or simply by C.

If a,~e ~, we sha l l wr i te

If

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ao ~ 2 (or ~1-a).

Le t

e , , 6 . . . . e , ~ '(1)

be a system of nonzero cyclic elements of the latt ice , ~0 , ~g ~) , and let

A~ c_ Bioa (Z=I,2 .... ) . If each A~ consis ts of pr eci sel y two elem ents and for

any a e A~ '9" eAJ" we hav e the con dit ion

ai o Ol/ Cl gi, o ~ ~ (~; ,~- -~.2 . . . . . r~) , (2)

then ~=(A~,~2,...,A ) wil l be calle d a complex define d by the element = and the

system of elements (i) . A complex defined by the null element and system (i)

is expressed by~ = ( e , , e 2 , . . . , e , , ) .

The set of all complexes defin ed by the element a and system (i) will b e

de no te d by /( ~o, (I)).

If # -- (2~,2~,,...,~) e/((b,(1)), we sha ll as sum e by de fi ni ti on that ~ =,p ~ ~i -~

By ~fi we shall denote the set of all complexe s ~ - ( 2 1 , / _ 7 z .... n ) , for which

there exist a,~,de ~s u ch that ~e K (a, (I) ) , pe. Kf ( , ( l ) ) , de oo { , #e K( i ( l ) ) an d

.o o jnAzo$+ . . . . .

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Proof. Let us take a syst em of elemen ts (i) that satis fies the condi tions

of the theorem, and denote by K the set of all complexes defined by cyclic ele-

ments of the lattice L and by system (I ). The conditions 1.4 and 1.5 of the

theorem define in K an associat ive algebraic operation, namely mult ipl icat ion of

complexes. Hence, in stea d of # ~ we shall wri te ~--~/o .

Let us show that $ e K (o, (I)) is a ri ght unit elem ent unde r the o per ati on of

mul tip lic ati on of complexes. Let ~ e/( (=, ( I )). For show ing that c~$= ~, it suf-

fices by virtue of the defini tion of a product of complexes to show that me m o o

and that

A o.nA oe (z./- . . . . .(This conditi on can be obta ined from (3) for fl-e, ~=cc.) The first a ssertio n is

evident . The second holds if Ag =8 (~ . However, from condit ion i .i of the

theorem , it foll ows that f or any ~=e 0 we h ave K Ca,(1))##. Fr om the d efi ni tio n of

a complex, it follows that if cc~ K(GoCIJ), then ~Z_~aogi, with ~ . Hence,

= o ~ ~ ~ for any = e C a n d any el. But in this case, also ~Z=gz. ~ ~ Thus, &

will be a right unit element.

Le t a e ~, ~ 0 , ~ e K (a, (13), c~le (~(1)) nd ~ ~ . The existen ce of such 0 and

~Ifollo ws fr om condition I. i. By using the conditi on 1.2 of the theor em and

the fact that O e Go ~ , we obtain, simila rly to the foregoing, the relatio n!

i.e., for = there exists a right invers e element.

It follows from the foregoing that K is a group under multi plic atio n of

complexes.

Let us prove that L IK) is isom orph ic to L. Let us wri te

{ , z l . e K

It follows from conditi on 1.3 that ~c consis ts of a sing le element. By also

taking into ac coun t co ndit ion I.i, we f ind that f is a map pin g of K onto ~.

Now let us define the mapping

9: Z , ( K ) - - ' - Z , ,

by writing for any subgroup H ~ K the formula

H = E H I ,

where H / is the set of images of all the elements of H under the mapping ~o. Le t

us show that is an iso mor phi sm of the lattices L(K) and ~. It follows from

the definit ion of ~ that if A ,2 ~e L( K) an d ~2 ~ , thenA~2~ ~. It remains to verify

that ~ is a one -to- one mapping. It follows from the defi nit ion of ~ that to any

subgroup /~c_K there corresponds a unique element of L. Let us show that any

eleme nt ge l is an image of a subgr oup of the group K. Indeed, let us wri te

40 2

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S i n c e a n y e l e m e n t o f /. i s a s u m o f a s e t o f c y c l i c e l e m e n t s , a n d ~ i s a m a p p i n go n t o 0 , i t f o l l o w s t h a t V # ~ . L e t u s s h o w t h a t V i s a s u b g r o u p o f K . L e t ~ ,~ e Va n d ~ # = ~, w i t h ~ = = , ~ = {, # ~- - ~ . I t f o l l o w s f r o m t h e d e f i n i t i o n o f a p r o d u c t

o f c o m p l e x e s t h a t ~ e a o [ ( h e r e w e a r e u s i n g t h e f a c t t h at a n y c o m p l e x i s d e f i n e d

b y a s i n g l e e l e m e n t; t h is fo l l o w s f r o m c o n d i t i o n 1 . 3) , i .e . , ~ + ~ = ~ % ~ = ~ + { .

H o w e v e r , a ~ , ~ J . H e n c e, ~ , a nd , t h e r e f o r e , #& V. H e n c e , V i s c l o s e d u n d e r

m u l t i p l i c a t i o n o f c o m p le x e s . F r o m t he p r o o f o f t h e e x i s t e n c e o f a r i g h t i n -

v e r s e e l e m e n t f o r z, i t f o l l o w s t h a t ~ - - ( - i )~ . H e n c e , i f ~ e ~ , t h e n o-le . T h u s ,

V i s a s u b g r o u p o f K . I t i s e v i d e n t t h a t V ~-- ~ .

F i n a l l y , l e t A a n d ~ b e d i s t i n c t s u b g r o u p s o f K a nd , f o r e x a m p l e , l e t J ~ ,

w i t h ~ E A , b u t ~c~2~. L e t u s a s s u m e t h a t A = T h e n $ ~ = / 7 S i nc eB # = Y, B ', w h e r e B ' = { { I ~ - ~ , / e ~ } , i t f o l lo w s th a t ~ B I. B y v i r t u e o f c o n d i t i o n

1 . 6, t h e r e e x i s t s a f i n i t e s e t o f c o m p l e x e s / ~ , , ~ ..... ~ , d e f i n e d b y e l e m e n t s i n B I

a n d b y s y s t e m ( i) s u c h t h a t

~ = P, A . . . # ~ .

I t f o l l o w s f ro m c o n d i t i o n 1 . 1 t h a tK ( ~ ( 1 ) )f o r ~ #0 c o n s i s t s o f p r e c i s e l y tw o c om -p l e x e s /~ a n d / ~ ; w e s h o w e d t h a t f l a n d f l a r e r e c i p r o c a l a s e l e m e n t s o f t h e g r oK, a n d , t h e r e f o r e , t h e y e i t h e r b e l o n g o r do n o t b e l o n g s i m u l t a n e o u s l y t o a ny ot h e s u b g r o u p s o f t h e g r o u p K . H e n c e , i t f o l l o w s t h a t i f ~ e K ( { , ( 1 ) ) ,h e r e { e ~ :t h e n ~ e B . H e n c e , ~ ,/ ~z .... m 6 ~ a n d ~ e ~ , w h i c h c o n t r a d i c t s t h e c h o i c e o f ~ .

H e n c e , A ~# /~ a n d ~ w i l l b e a o n e - t o - o n e m a p p i n g . T h i s c o m p l e t e s t h e p r o o f of

t h e t h e o r e m .

~ 2

I n t hi s s e c t i o n w e s h a l l s h o w t h a t a l a t t i c e o f s u b g r o u p s o f a n o n c o m m u t a -

t i v e g r o u p G i n w h i c h a n y t w o e l e m e n t s g e n e r a t e a f r e e g r o u p w i l l s a t i s f y t h ec o n d i t i o n s o f T h e o r e m i . S u c h g ro u p s w i l l b e c a l l e d 2 - f r e e g r o u ps ( w i th r e g a r d

t o th e e x i s t e n c e o f s u c h n o n l o c a l l y f r e e g r o u p s , s e e [ 8] ).

I n t h e L e r m na s 1 - 5 f o r m u l a t e d b e l o w, w e a r e c o n s i d e r i n g s u b g r o u p s a n d e l e -

m e n t s o f a 2 - f r e e g r o u p G .

L E M M A1 . I f tQ.1INt~J-=-E,t h e n {al o{ ~J w i l l c o n s i s t o n l y o f t h e f o l l o w i n gs u b g r o u p s [ a { J , { a { - ' t ,{ a " { ] , { a - ' { - ' } .

T he p r o o f i s e a s y t o o b t a i n b y v i r t u e o f t h e f a c t t h a t ~, a n d ~ a r e f r e e l yg e n e r a t i n g t h e s u b g r o u p ta , d } .

LEM~~ 2 . I f { ~ ) n { a J = E , ] # t , a , X '~ O, 5 ~ 0 ,

t ] ' ~ a : J t ~ ~ 6 t n { ~ t o [~ ~} ?~ ~ , ( 4 )t h e n ~ = S

P r o o f. I f { ~ = = } N ~ 9 ~ s J - - E , t h en , b y v i r t u e o f L e m m a l , i t f o l l o w s f r o m

( 4) t h a t f o r c e r t a i n 8 = + I, 2~ = - +4 a n d i n t e g e r { w e h a v e

( f ~ a ~ ) ( ]e a ' ~ ,l ~= ]

403.

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This equation is possible only for 8=I,%=-/,.~=S, by virtue of the fact thatand c are freely generating the subgroup[~,=}.

I f I ~ } ~ { ~ s } ~ ~ , t h e n { ~ x ~ J a n d {~} will be contained in the same ~ z ~ ~ $ ~ ~ . .

.cyclzc subgroup s. Ther efor e, ~ a ~ a ~ ~ a ~ ~ , whzch zs possible only forK= d, ~= 8. This completes the proof of Lemma 2.

In the same way we can prove the following two lemmas.

LEM M A 3. I f { q ~ n { ~ = E , ~ ( an d { ~ } ~ { ~ } ~ { ~ o { ~ } ~ ~ , t hen { ~ } =

LEM MA 4. If {/r}O{uT} = ~ an d l K~ } n { ~ } #~ , t hen for any K~ d we have

~ . ~ 5. Le~ . ~ g , { = ~ . ~ ,. = ~ e ~ C ~ = ~ . ~ ..... ~ ) , w it h { { } ~ { 5 } a n d

{ur/,}~: 1%.}for 6+j . If {zr~-n JpJ= {~rJnJa}={~r}n{6'} .={~-}n/c} ={~ }h -[~ '}= f ,and for any pair g,7' ( 4 / ' - - 4 2 . . . . . re)where exist 2 , , , . = + . / (K= ,z,3) such that

then c = ~.

Proof. It is evi den t that any of the equa tio ns ~4'c--i, z~a ~ i and (by vir-&

rue of Lemma 4) any of the relat ions{~t'cJnluf]4E and {6 28 }O {~ }+ f can hold for

not more than one value of ~ for any ~,=il ,l~-~il. Wit hou t loss of generality,

it can be assum ed that if there exist such values of ~, they will be fou nd among

the nu mb er s I, 2, 3, and 4.

If g>f, then (as before) it follows from Lemma 4 that for not more than

one 7'-/(~) we ca n h av e

1 %t n { '5"dJ + e .for any %--I, ;~z +- L

Since ~m5 and/#/Cb), i t fol lows that for any ~=I (K=42 .3),

I n view of this we find, by virtue of Lemma i and of (5), that for any pair ~,j

whe re 6=F.F .... y, /~/( 6) , we ha ve at least one e qua tio n of the f orm

(~'O) @ d ' ~ ) @z~- (Ctrt'a)SJ~8') 8s , (6)

where ~ ---- i [~ = Y,2,&~,).Let ~ be fixed, ~), and let / run thro ugh the values from i to/a, with

/~ /f i ) . Since ~>~258, and there exist precisely 256 eight-sets of the form

(~1, ~,~z ,~, 7~3 ,6, @~, 8~) , it follow s that there exist Y'~,/z '/ I ~ , such that

for the same set of values of ~K,~.

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L e t u s s h o w t h a t ( 7) a n d ( 8) c a n s i m u l t a n e o u s l y h o l d o n l y f o r 8 5 = I. I n -

d e e d, l e t 8 = - / a n d, f o r e x a m p l e , l e t 8 - ~ . F r o m ( 7 ) a n d (8 ) w e o b t a i n

F r o m t h e s e e q u a t i o n s w e o b t a i n

H o w e v e r , t h i s e q u a t i o n c a n n o t h o l d , s i n c e u n d e r o u r a s s u m p t i o n w e h a v e { 6Z ' c ~ o I u/~

= E , ~ ' C ~ ~, u / ~ f ,a nd t h e s u b g r o u p ~ J ? C , ~ i s a fr e e g r o u p .

I n t h e s a m e w a y w e c a n s e e t h a t t h e c a s e @ ~ =- 4 8 = - ~ i s a l s o u n f e a s i b l e .

T h u s , ~ = 4.

F o r ~s--- / w e o b t a i n f r o m ( 7) a n d ( 8) t h e f o r m u l a

I f ~ , t h i s e q u a t i o n w i l l b e v a l i d ( b y v i r t u e o f t h e f a c t t h a t {u/} { {} === a n d

{ ~ ,~ } i s a f r e e g r o u p . ) o n l y f o r B z = - ! a n d A ~ = - ~ . I f ~ = 4 , i t w i l l h o l d o n l y f o r

IzSz = 9 # . I n e i t h e r c a s e ( b y v i r t u e o f t h e f a c t t h a t 8 s = I ) i t f o l l o w s f r o m ( 7)

t h a t f o r a n y i ~ F w e m u s t h a v e

( 0 " ~~ ' c ) 8 , f - ,= ( E ; ' a ) ~ . ( 9 )

B y a s s i g n i n g t o L t h e v a l u e s f r o m 5 t o m , w e f i n d t h a t f o r c e r t a i n ~ j. ~

f z ' w e h a v e

c) = , ( l O )

= ( l l )

f o r t h e s a m e s e t o f v a l u e s o f 2 , , , ~ , 1 ~ , ~ 3 . . .

L e t us c o n s i d e r a l l p o s s i b l e c o m b i n a t i o n s o f th e v a l u e s o f ~ , ~ , ~ , ~ .

I f I ~ = % ~ , 8 1 =~ , ~ = /, i t f o l l o w s f r o m ( i 0) t h a t c = d F .

I f ~ , = $~, ~ , = / , ~ - - / , w e o b t a i n f r o m ( I0 ) a n d ( Ii ) t h e e q u a t i o n

i . e . , f o r a n y S ~ O w e h a v e

/ I S , 2 - / / . f s = = ~ Z - " .

T h i s e q u a t i o n c a n n o t h o l d i n a f r e e g r o u p ~ a ~ .

I n t h e s a m e w a y w e c a n s e e t h a t t h e c a s e % 1= ~j , 01 = - 4 , @ 3 = / i s a l s o i m -

p o s s i b l e .

I f 2~ = ~j , G I = - I , ~ - - - I , w e f i n d ( s i m i l a r l y t o t h e f o r e g o i n g ) f r o m ( I0 )

4 0 5

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and ( I i ) that for an s~ Ow e have

j - s =

This equat ion cannot hold for ~y~/, s ince I ~J n{~ J= Ean d tO',(J is a free group.For ~=( i t fo l lows f rom (I0) that (~ i= ~ , ~= -4 ~- ~- / )c =a , i .e ., c- -- -=~.

In the same way we can co nside r the case that ~1~%j . This co mplete s the

proof of Lemma 5.

COROLLARY. If ~= ! under the condit ions of Lemma 5, then c =~ .

Let $ be a group. Let us recal l that cycli c e lemen ts of /, [G) coin cide w ith

cycl ic subgroups of $. Instead of cycl ic elements of L(G), we can therefore re-

fer to cyclic subgr oups of $ and vice versa. Let

t ~ , t , ~ z } , ' ' " t ~ } (12)

be a sys tem of nonuni ty cyclic subgroups of the group G, {a ~E . I f ~ cons i s t s

of sub gro ups {~= } a nd {~ffa}, wi th {~i=}~{~'I~}, (~ ~-~.2 .... n) , we can easi ly see

that

= (4 , A z . . . . . A )e K (taJ.

Such a complex wil l be denoted by i< ~,(12)). It is natural to assume thatK C~', C~2 ~)= ~ .

In general, K (~, (12)) can be specified not only by a system of subgroups(12) and the element a, but also by a system of generatrices selected in the

subgr oups (12). However, if K(a,(12)) xists in a tors ion- free group, it wil l be

unique ly dete rmined by system (12) and the element ~. Since in a torsion-fr ee

~ a~ if and only i f ~- ~ = ~z , we obtain the fol lowingroup we have ~ } = -~ "~

lemma.

LEM MA 6. If G is a tors ion- free group and for any ~,/~ a e ~, we have

j ~ =-i, i t : fo l lows that for any sys tem (12) and any =e G there exi s t s a unique

complex K ( a, (12)).

LE MM A 7. Und er the con dit ion s o f Lermma 6, if = ~- K (~z, 12)), ~=/~ (~,(12)) ,the n ~ , ~ K (= (, (12))~

For the proof i t suffices to use the defini t ion of ~p.

THEOR EM 2. A lat t ice of subgroups of a 2-free nonco mmuta t ive group sat is-

fies the condit ions of Theorem I.

Proof. Let G be a 2-free nonc ommut at iv e group. It is evident that L(G) is

a complete lattice..

As we noted above, the set of cycl ic elements of L(G) coincides w ith the set

of cyclic subgro ups of $. Hence, any eleme nt of L (G)will be a sum of a set of

cycl ic elements.

L et ~ G , ~ g ! (K=~,2 . . . . p) a nd I ~ I ~ = fo rK ~. Such a se t of e lementsof $ exists for any natura l p, s ince ~ contains a free nonco mmuta t ive subgroup

406

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(see, for examp le, [2]). Let us wri te ~ui-~u , Z~/. ~O, E~i=~} (L,=4,...,,'n)witha~g ~ ; ~ for i~j. We obtain a system of element s of the lattice I~):

. . . , e , , . , , . . . . . . ee, , . . (13)

Let us prove that for m~ f an d n~m2ES, system of elements (13) satisfies

conditions i.i-1.6 of Theorem i.

Cond itio n i.i. The conditions of Lemm a 6 are satisfie d in a 2-free group.

Ther efore , the co mple xes ~ N (a,(la)), ~'= K (,Cla)) exist and are un iqu ely deter-

mined, with ~ i for l a 3 ~ . Let us show that other complexes defi ned by {a} and

syste m ( 13) do not exist. Let {~ J# E and let

,A"= . . . . . . . . ) (

For at least one ~ we hav e{ c~ }n ~K }= E. By the defini tion of a complex we haveI I I I

AKin--{~K~} o ~ } By us in g Le mm a i, we f ind that AK~ can cont ain only some of the

subgroups

Let , for e xample , A IK~ { ~ G ~ } , whe re ~=-+i .8=+-I , but the values of and 8 are

fixed. If A II ~' a 8'j 9 {~Kj ~, then by the definition of a complex,

By using the rela tion ~x~ ~ and Lemm a 2, we obtain ~=8.

Let {=} ~{~ K}= [=J n{& } = E K~$ , As we just showed, A n~6 can contain only

subgroups of the form { ~ ~81, an d Air j can con tain onl y su bgroups of the form

{$~~7 = ~' wh er e @ and 8, do not ,depend on L and j. By de fi ni ti on of a comp lex ,

we ob ta in fo r any Aj = 4,2 ....

where ~ ,/%~= 1. Fro m these rela tions we find, by virt ue of the corollar y ofLe mm a 5, tha t @ = 8~.

Fina lly, let I]~} ~ {=J = E, {~zJ ~ J ~ E Sin ce ~= and ~ g ene rat e a free

group, there exists an ~e &s uc h that ~ze {~ j and~ e{~ ]. On the other hand,

A" c ~ J o ~ J . Hence , for any subgro up ~ ~A "J we hav e {FJ~--_{~J . For int.egers~t, "~z' and s~we ha ve ~=~s,, f=~s~, ~ .=/~% . ByZ~virtue of AK~:~{~ @~ and of thedefinition of a complex, we obtain

By subs titu ting into this formula the value s as =~ 8~', ~=~s~, ~7~j= h% and usingLemma 3, we find that {br}={~zj~jor {~={~z; ~ j .

Thus, we ha ve shown that for any compl ex ~// there e xists a value of 8

( @ - ior@---l) that does not d epend on ~ and j and such that any ~j can contai n

only subgroup s of the form {~;xGs}. But acco rdin g to the defin ition of a com-

plex, A~j contains pr ecis ely two cyclic element s belo ngin g to /~G). Therefore,A" ~will consist of the subgroup s {~z/=O and {~ ;=~ } . But in this case for ~=

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the complex ~ will coincide wit h ~ , and for @---lit will coincide wi th =i . Thus

condition I. i of Theorem i will be satis fied for syste m (13).

Condition 1.2 will now assume the form!

! 1

A,s e~7~t~,~ ~ j J i. e. conditio n 1.2 is sa tis fie d.

C o n d i t i o n 1 . 3 . L e t ~ e K ({aJ , ( la) ) , f leK(l{J , ( la) ) ,

f = (~,,, & ,. .. , B,~, %,,...,~pm).As we showed above (condition i. i), it follows from ~ K(l ~J, (la )) that ~ = K

(~,(13)) or ~-- K ~ a-i,(la)) Wi th ou t loss of gene ral ity , it can t her efo re b e as-

sumed, for exa mple , that ~ = K (=, (IS)), fl = K (~,(la)) Le t us as su me that ~=/~ .

This signifies that for any 4~ K_ zp an d 4_~g~ra we have ~~=~g. Since pmF ,

there exist two subgroup s a mong {~}, for example, ~i~ and {~il' such tha t

tg,} n{ a ) = f~,} n f~3= to~} n ta} - f,} n l~ } "=e . (14)

Since ioz. )~2~=. and ~ = ~ z ; , it follows that {~/.F}e~ . In vie w of this it fol-~jlows fro m the ~ef in iti on of a co mp lex that for a ny f,~--/,2 .... z,

. . . . . . . ,,t M

Fro m the cor olla ry o f Lemma 5 it follows that a--~, i.e.,{a} = l~}.

Con dit ion 1.4. Suppo se that the complex es ~ and ,8 wer e de fine d as above,

an d t ha t # e K ({d}, rla)),

2 = ( 4 , , 4 ~ , . . . , 2 , m , ~ , , . .. . ~ p ,. ,, ),

here we can assume that

As in the verif icati on of condition 1.3, it can be assumed that conditions (14)

hold, and, moreover , that {~} ol d) = ~.

Let us assume that ~e~#. It then follows fr om the definition of ~# (con-

dition (3)) that for any /~,/=1 ,2 .... m ,

In going ov er to the elemen ts of the sets _~, 2~., Ali, we find that for any pair

@" , wh er e i,/=,= .... m, the re ex is t ~b~ -=-* ! (~=!,2, ) su ch tha t

s ) ; . { gz,,By v i r t u e o f Lemma 5 , i t h e n c e fo l l o w s th a t d =a ~ . T h u s , we h a v e p ro v e d th a t~ / ~

consists of not more than one complex. However, it follows from Lemma 7 that

~/~ffK (=~,(I~). Thus, c ondit ion 1.4 is satisfied. It henc e fo llows that by de-

noting with K a set of complexes specifi ed b y cyclic elements of /. (G)and by

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s y s t e m ( 13 ), w e t h u s d e f i n e i n K a n a l g e b r a i c o p e r a t i o n , n a m e l y , m u l t i p l i c a t i o n

o f c o m p l e x e s .

I n c h e c k i n g c o n d i t i o n i . i , w e h a v e p r o v e d t h a t K ( {a ~, (l a) )i s e x h a u s t e d b y

t h e c o m p l e x e s ~ = K ( ~, (1 8) ) a n d ~ / - - ~ (~'/,(Is)), w h e r e a s i n c h e c k i n g c o n d i t i o n

1 . 3 , w e s h o w e d t h a t i f ~ = K (~,(IS)), ~ - - - K (~ ,( 13 )) n d ~ = ~ , t h e n ~ = ~ . H e n c e , t h e

m a p p i n g ~, a c c o r d i n g t o t h e r u l e

w h e r e = = K ( ~, (Is)), i s a o n e - t o - o n e m a p p i n g o f K i n t o G . M o r e o v e r , f r o m t h e

p r o o f o f c o n d i t i o n 1 . 4 ( i .e . , f r o m t h e e q u a t i o n ~ = a ~ ) i t f o l l o w s t h a t ( ~ ) ~- - ~ P p~ ,

i . e. , ~ i s a n i s o m o r p h i s m o f K a n d $ u n d e r t h e o p e r a t i o n o f m u l t i p l i c a t i o n o f

c o m p l e x e s a n d t h e g r o u p o p e r a t i o n i n $ . I n v i e w o f t h is i t is e a s y t o s ee t h a t

c o n d i t i o n s 1 . 5 a n d 1 . 6 h o l d f o r s y s t e m ( 13 ). T h i s c o m p l e t e s t h e p r o o f o f T h e o -

r e m 2 .

3

L e t e , , e ; . .. . o p ~ C ( L ) , w i t h e ~e ~ O f o r K ~ ; S x . " ~ ( b ~ 4 2 ..... n), ~.i ,~e Kj f o r 6~ 2 .

T h e s y s t e m o f c y c l i c e l e m e n t s

e , , , % . . . . . e , ~ , 6 ~ . . . . . e z ~ . . . . . e p ~ ( 1 5 )

f o r p > . 5 , n z ~ 2 F 8 i s c a l l e d t h e b a s i c s y s t e m o f t h e l a t t i c e L.

L e t u s n o t e , f o r e x a m p l e , t h a t f o r p ~ E , n ~ f F $ , s y s t e m o f e l e m e n t s ( 1 3 )

c o n s t r u c t e d a t t h e b e g i n n i n g o f t h e p r o o f o f T h e o r e m 2 w i l l b e a b a s i c s y s t e m

of t he l a t t i c e L CG~ .

F o l l o w i n g [ 7] , a l a t t i c e g w i l l b e c a l l e d a t o r s i o n - f r e e l a t t i c e i f t h e

i n t e r v a l ~, /0 i s f i n i t e f o r a n y ~ e L , u ~ O .

T H E O R E M 3. A g r o u p $ i s a f r e e g r o u p of r a n k ~ 7 2 i f a n d o n l y i f L ( G )

s a t i s f i e s t h e f o l l o w i n g c o n d i t i o n s :

3 . 1. 6 ( G ) i s n o t a d i s t r i b u t i v e t o r s i o n - f r e e l a t t i c e.

3 .2 . I f {a } a n d { {} a r e n o t s u b g r o u p s o f t h e s a m e c y c l i c s u b g r o u p a n d

{d} e{ a} o { ~ } , t h e n Id J n t ~ } = ~ / } n l ~ = E ,

3 .3 . T h e r e e x i s t s a t l e a s t o n e b a s i c s y s t e m ( 1 5 ) o f t h e l a t t i c e L ( ~ )

3 .4 . T h e r e e x i s t s a s e t ~ o f c y c l i c e l e m e n t s o f t h e l a t t i c e L ( O ] s u c h t h a t

ES=~, a n d i f { ~ i } ~ } E S ( L = / , 2 . . . . .s),

t h e n

( ' " ( ( ~ , ~ z ) ~ 3 ) ' " ) ~ s ~ g " (16)

P r o o f . N e c e s s i t y . L e t ~ b e a n o n c o m m u t a t i v e f r e e g r o u p. I t i s e a s y t o

s e e t h a t c o n d i t i o n s 3 . 1, 3 . 2 , a n d 3 . 3 a r e s a t i s f i e d i n L ( G ) . L e t u s s h o w t h a t

c o n d i t i o n 3 . 4 a l s o h o ld s .

A s S w e s h a l l t a k e t h e s e t o f a l l c y c l i c g r o u p s w h o s e g e n e r a t r i c e s a r e

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elements of a system of free generatrices of the group ~. Then ~ $ = G.

It follows from the proof of Theorem 2 that the set of all complexes de-

fined by elements of ~ (Z (~)) and basic syste m (15) form a group K under multi -

plica tion of complexes, and that the mapp ing ~ define d at the end of the proo f

of Theorem 2 is an isomorphism of K and $.

If ~ ~ K f{=gj,(Is)) the n ~ = - K (=~, (Is)) or ~l =K(a-~,(m)), s it follows

from the proof of Theo rem 2. Therefore, ~- -= ~ or ~ = ~-1 and

. . . .~,,~, ~ ~ ,~ [. ~

where ~=+t. I f {=~;~I~.} ~-~ and {~}+t[7 +,~ , then =f'=zSz...o:s ~ Ib y vi rt ue of

the definit ion of $. But in this case, also, % ~ ""~s ~ 5 , i .e., f ormula (16)

holds. Thus, condition 3.4 is satisfied.

Suffi cienc y ~ Let conditions 3.1-3.4 be sati sfie d in L(&). By virtu e of

conditi on 3.3, there exists in L(~)a basic sy stem (15):.

By virtue of conditi on 3. I, $ is a nonc ycli c torsio n-free group. It fol-

lows from the condit ion 3.2 that ~ -~ -~ for any ~, $6 ~, ~/ . Indeed, i f

~ ' ~ = O "~, then [ ~ } ~ { 9 ~ = { ~ ~ ~ and at the same time {~a) e { ~ o {aJ , whi ch con-

tradicts condit ion 3.2. By virtue of Lemma 6, there exists for any a e S a com-

plex ~ = K (a,(Is)). I f ~ = K(~,(Is)) , then by vir tue of Lem ma 7, we have ~ s K

(o~,(IS)). Hence, it follow s that if ~ = K (ag,(Is)) (~=/,2,...,~), he n

( . . . C~,~ ~% . . ) % ~ K ~,a~ . . a s , ( l ~ ) ) .

Let us assume that $ is not a free group. Since G is a nonc ycli c torsion-

free group, it follows that for any set S of cycl ic eleme nts of D C$) such that

Z S = & , there exist al,a .... a e O s u c h that {og}~ i~ Je S, i~J#{~.~j and Q~az... ~s = ~. In this case ,

( . . . ( ~ , % ) % . . . ) % ~ K ~ a, a~ . . . a s , ( 1 ~ )) = K U . ( 15 ) )- - -~

and condit ion 3.4 does not hold in L(G). Hence, $ will be a free group. Thiscompletes the proof of Theorem 3.

THEO REM 4. If $ is a 2-free group, ~e L (~7, then ~ ~ $ if and only if for

any cycl ic elements ~ ,~ ,c~ L ~7 such that O ~ ' = g , ~ ,

the condition c~2~ is satisfied.

Proof. Let ~=la1~, ~= I~ }. If c e-(ao~f) o= , then CE~o~, w h e r e ~ e G o ~ .Since =~=O, i t fol lows from Lemma i that ~= ~@ ~z } fo r~ i ,~ = I. But in this

case c~a=O, and, by virtue o f Lemm a i, there exist 8~,8 #=i4 such thatO={(~z~4~)SJ~Z84}. If ~=-f then C---~-~Q,~+@~ and hence ce(a.a).~. Hence,8S= / and c={~18t~,~f4}. Let us assume that ~7~I and ~ = ~ . Then

x = t a , ~), a ,a , o~ ~z, ~eco c,

but ~ (a,(~o{)),a. Hence, if ~,~i, then 4+ 0 4 . Therefore, C=la, 48La; } or

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c={a~ '~al} . Th is comple te s the p roof o f Theorem 4.

A la t t ice L that sa t isf ies the condit ions of Theorems I and 3 is ca l led a

free-group la t t ice .

THEO REM 5. A la t t ice L is isom orphi c to a la t t ice of subgroups of a f ree

group of rank z~2, if and only if i t is a f ree-group la t t ice.

Proof. Let L be a f ree-gro up la t t ice . Then the condit io ns of Theo rem i

wil l be sa t isf i ed in L, and hence L will be i somor phic to a la t t ice of sub-

groups of a group G. Since L sa tisf ies the condit ions of Theor em 3, i t fol lows

that G is a free grou p of rank z~.

Conversely, a la t t ice of subgroups of a f ree group of rank ~ Z is a f ree-

group la t t ice by vir tue of Theorems 2 and 3. This completes the proo f of the

theorem.

Elements of a f ree-group la t t ice L that sa t isfy the condit ions o f Theor em

4 are ca lled normal e lements of L.

Let us specify a c lass F of la t t ices. More precise ly, we shall inc lude inF a l l the f ree-gr oup la t t ices and a l l the la t t ices that are intervals~/F, where

is the unit e lemen t and ~ is a norma l e lemen t of a f ree-group la t t ice.

I t fol lows f rom Theorem 5 that there exists an isomorphic mapping of any

free-group la t t ice L into a la t t ice of subgroups of a f ree group G. As i t fol-

lows f rom Theor em 4, in this case normal e lements of ~ are map ped into norma l

divisors of ~ . I t a lso follows f rom Theorems 4 and 5 that a la t t ice L(G} of any

free group G of rank ~Z is a f ree-gr oup la t t ice and that normal divisors of

are norma l e lemen ts of L(&). Thus, the c lass ~ wil l consist only of la t t ices

that are isomorphic to the la t t ices L(G/H), wher e ~ is a free group of rank ~2 ,and H is any of i ts norma l divisors. Hence, we obta in the following theorem.

THEO REM 6. A la t t ice L is isomo rphic to a la t t ice of subgroups of a group

if and only if L is isomorph ic to a la t t ice of c lass F.

4

THEORE M 7. Any la t t ice i somo rphism of a noncommu ta t ive 2 - f r ee g roup is in -

duced by one (and only One) group isomorphism.

Proof . Let G be a 2-free nonco mmut ativ e group, and le t ~ be its la t t ice

isomorphism, I t fol lows f rom Theor em 3 that G ~ is a lso a 2-free noncon~nuta tivegroup.

In L(G) there exists a basic syste m of e lements

I t is evident that

e, ,5 . . . . e~. (17 )

e,~, e f e~ (18), .

is a basic syst em of e lements of L(G~').

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Let K and /~I be the sets of all compl exes of L(G) and L(G~), def ine d by sys-_ _

terns of elements (17) and (18). Since the concepts of a comple x and of a produ ct

of complexes are define d in terms of lattice theory, there exists betwee n K and

K a one -to -on e cor res pon den ce ~ such th at if ~ e K ({=}, (17)), then

As we noted at the end of the proof of Theo rem 2, a corre spond ence ~ under

which for any = Gw e have

a f' = K (a,(17))

will be an isomorph ism of K and ~ with respect to multiplic ation of complexes

and the group opera tion in ~.

Similar ly, if f or any s,e ~Y,

( ~" Ca, , (,8)))~=~,,

then fz will be an is omo rph ism o f the gro ups K I and G ~. Now it is easy to see

that { # ~ ~ z is an isomorph ic mapp ing of ~ into ~Y. In this case, if s~--al,

then{a}-~-{=1], i.e.,{~} = {~ {] Thus, there exists an iso mor phi sm { of the

groups ~ and &~ that induces ~.

Finally, if { and {, are isomorphisms that induce ~, a, {e ~, {~]n {~] = E,

then {~ ] ~- {~{~t}={af'~').- Hence, follow s that a { = ~ {' , i.e., { = ~ . This com-

pletes the proof of Theorem 7.

LITERATURE CITED

I. M. Suzuki, Structure of a Group and Structure of a Lattice of Its Subgroups

[Russian translation], IL, Moscow (1960).

2. M. I. Kar gap olov and Yu. I. Merzlyakov , Foundat ions of Group Theory [in

Russian], Nauka, Moscow (1972).

3. S. A. Anishchen ko, "Rep resen tati on of mod ular lattices by lattices of sub-

group s," Matem. Zap. Kras noy ars kog o Gos. Ped. In-ta, No. i, 1-21 (1965).

4. L. E. Sadovskii, "Lattice isomorphisms of free groups and free products,"

Matem. Sb., 14, No. 1-2, 155-173 (1944).

5. B. I. Plotkin, "Problems of the theory of torsion-free groups," Ukr. Matem.

Zh., 8, No. 3, 325-329 (1956).

6. O. Ore, "Structures and group theory, II," Duke Math. J., No. 4, 247-269

(1938).7. P. G. Kontorovich and B. I. Plotkin, "Lattices with additive basis," Matem.

Sb., 3__5, 187- 192 (1954) .

8. G. Baumslag, "On generalized free products," Math. Z., 78, 423-438 (1962).