amalgamation bases for nil-2 groups

Algebra Universalis, 16 (1983) 4 7 - 6 2 0003-5240/83/001047-16501.50+0.20/0 �9 1983 Birkhauser Verlag, Basel

Amalgamation bases for nil-2 groups

D. SARACINO

Let 7"2 denote the first-order theory of nil-2 groups (i.e., groups nilpotent of class at most 2). One troublesome aspect of the model-theoretic investigation of T2 is that 7"2 does not have the amalgamation property. In this note we characterize the amalgamation bases for 7"2 and obtain some further results about them (w Along the way we also determine conditions for the adjoinability of roots for elements of a nil-2 group in a nil-2 extension, and for the embeddability of a nil-2 group in a divisible nil-2 extension.

Our results will be applied in [5] to the construction of 2 so elementarily inequivalent quantifier-eliminable nil-2 groups of exponent 4 (necessarily N0- categorical).

1. Notation and preliminary results

Let G be a group, let x and y be elements of G, and let A and B be subgroups of G. Then Ix, y] denotes the commutator x - ly - lxy , and [ A , B ] denotes the subgroup of G generated by all commutators [a, b], where a ~ A and b ~ B. In particular, the commutator subgroup of G is G ' = [G, G].

We will sometimes denote the quotient group G/G' by G; then, for g~ G, denotes the class of g in G. We write Z(G) for the center of G.

A group G is nil-2 iff [x, [y, z]] = 1 for all x, y, z in G, i.e., iff G ' _ Z(G). We will use the following rules for working with commutators in nil-2 groups.

PROPOSITION 1.1. [6, p. 177]. I f G is nil-2 then for all x, y, z ~ G and all integers m and n we have

(i) Ix, yz ]= [x , y][x, z], [xy, z ]= [x , z][y, z], and [x% y'~] = Ix, y ]"" ; (ii) (xy) ~ = x'~y"[y, x] '~('~-x)/2.

Our principal tool will be the second nilpotent product of groups. If A, B are

Presented by B. J6nsson. Received January 29, 1981. Accepted for publication in final form September 30, 1981.

47

48 D. SARACINO ALGEBRA UNIV.

nil-2 and F is the free product A * B, then the second nilpotent product of A and

B is

A (~) B = F/[F, [/7, F]].

A (2)B has subgroups naturally isomorphic to A and B, and if we identify A and B with these subgroups then every e lement of A (~)B has a unique expression in the form o~/33,, where a ~ A, /3 e 13, and 31 s [A,/3]. A theorem of T. MacHenry states that the subgroup [A,B] of A ( ~ B is isomorphic to the tensor product

A |

P R O P O S I T I O N 1.2 [3]. There is an isomorphism [A, B]---> fi, | B such that [a, b ]~-> fi | b.

At several points we shall consider second nilpotent products

A a ( ~ A 2 Q ' " "(2)Am of more than two groups. These are defined as for two

factors, starting with the free product A1 * A2 * �9 �9 �9 * Am. One last bit of notation: We write G ~ q~ to indicate that the s ta tement �9 is

true in the group G.

2 . R o o t a d ] u n c t i o n

In order to characterize the amalgamat ion bases for nil-2 groups, we will need to know under what conditions an element of a nil-2 group G has an nth root in a nil-2 extension of G. More generally, given a positive integer m, an m-tuple

n = ( n l , . . . , n~) of positive integers, and an m- tup le g = (g~ . . . . ,.gin) of elements of G, we determine necessary and sufficient conditions for the existence of a nil-2

extension of G containing an nith root for gi (1--<j-< rn). Given G; rn, n, and g as indicated, consider the following condition: for every

m x m array {qj} of integers such that nic~ i=nsq~ for all i,], and for all

Yl, - - -, ym6 G,

y~,-=- g~,i(mod G ' ) ~ [yj, gi] = 1. i ~=1 j = l

I t is easy to see that there is an infinite conjunction 4~m,.(vl . . . . , vm) of formulas of the first-order language of group theory (involving free variables v l , . . . , vm) such that for any group G and any m-tuple g of elements of G, the above condition holds iff G~r (r has one conjunct corresponding to each

Vol. 16, 1983 Amalgamation bases for nil-2 groups 49

choice of an array {qi} and a bound on the number of factors in an expression for an element of G ' as a product of commutators.)

T H E O R E M 2.1. Let G be a nil-2 group, let m > 0 , let n be an m-tuple of positive integers, and let g be an m-tuple of elements of G. Then there exists a nil-2 extension H of G containing an nith root for gi (1 <- j <-- m) iff G ~ q)mj,(g)-

Proof. First suppose that H is a nit-2 extension of G containing elements rj such that r~ = gi (1-< j-< m), and suppose we have y~',-~ I] g~(mod G') in G, with n~qj = niq~. Then in H

g,', ~ = I [ [r rj] i =I i i " i ~ i

=1-I ({r176 d i r t ', rJ) =l-I ([r~, ~]",~,,-",~,,) =1-I It,, ~j]o= 1. i < i i < i

Conversely, suppose that G ~ ~,~.~ (g). To adjoin roots for the g~'s we will use the group C = (x~)(~)---(~)(x~), the second nilpotent product of m infinite cyclic groups. Every element of C has an expression x~ . . . . x ~ I-[,<i [x~, xj]k,, where the a~'s and ~j 's are uniquely determined integers.

In G(~)C, let N be the smallest normal subgroup containing g~x]-", . . . . . g~xT~ "~. We want to show that N N G = { 1 } , for then ( G ( ~ C ) / N provides us with the desired extension H of G. So suppose that

s.

i = 1 -

(i)

where each s i is a positive integer, each eik= 4-1, each bik ~ G, and Xi~ = 1-I~%~ x~,~ for some integers aik,. (The left side represents a general element of N since G ( ~ C is nil-2.) Eq. (1) becomes

( ~ [(b~kX~k) -1, (gjxF')-~'~](gjxF")~'~) = go,

(2)

where ~. = ~k eik- If we write the left side in the form a/3% with a ~ G,/3 ~ C, T [G, C], the /3-fac tor is x~ ~,~ . . . . x~ ~ '~- z, where z ~ C' , so since/3 = 1 by uniqueness, we have -njtj = 0 for each j, so t i = 0 for each ]- Again by uniqueness, (2)

50 D . SARACINO

yields

and

Feeding in the value of Xjk and rearranging, these become

1 =I-I ([x~. xi]-.,z~:,~,+.,z~,:,~o in C, i<i

and

]),(, =l~j ([(~b~)-'~g:[Z~':%xiJ)in[G,C],

A L G E B R A UNIV.

(3)

(4)

(5)

Y}-"~I'I g~:"| in (~ | i

for each j, so

y ~ - - I I g~(mod G') i

where 57 means 57~=1 or Y~}=I depending on whether eik or elk is involved in the sum. ff we now define Yi e G by yj = 1-Ik b~ ~, and define cii by qi = - ~ ema~gj, then Eq. (5) becomes

If we apply MacHenry's isomorphism 1.2 to this equation, and then the standard isomorphism G | C = G ~ ((~i 07~->) ~ (~j (G (~ 07~>), we get


for each ]. Also, by Eq. (4) and our choice of cli's, thqi = njq~ for all i, 1. Thus since G ~ , , , . ( g ) , we get l'Ij [Yj, g i ] = l . By Eq. (3) and our choice of yi's, this says go = 1, as desired. []

The condition ~,,, . becomes relatively simple when m = 1:

C O R O L L A R Y 2.2. Let G be a nil-2 group, let g ~ G, and let n > O. Then there

exists a nil-2 extension H of G containing an nth root for g ill for every integer c,

G ~Vy(y ~ --- gC(mod G') ~ [y, g] = 1).

We remark that Wilfrid Hodges has (independently, but by methods very similar to ours) described a "recursive Nullstellensatz" for nil-2 groups and obtained 2.2 as an illustration.

In view of Theorem 2.1 it is natural to guess that a nil-2 group G is embeddable in a divisible nil-2 group if[

G ~ ~m,. (g) for all m > 0 and all m-tuples n and g. (6)

We now show that this guess is correct, and that (6) is equivalent to some simpler conditions.

T H E O R E M 2.3. Let G be a nil-2 group. The following are equivalent:

(i) G can be embedded in a divisible nil-2 group

(ii) G ~ qbl,,(g) for all n > 0 and all g e G (iii) Every element of G that has finite order rood G' is in Z(G) (iv) G ~r (g) for all positive m and all m-tuples n and g.

Proof. We show that (i) ~ (ii) ~ (iii) ~ (iv) ~ (i). (i) ~ (ii): This follows from Theorem 2.1. (ii) ~ (iii): (ii) says that G ~ Vy (y" = gC (rood G') ~ [y, g] = 1) for all c, all n > 0,

and all g ~ G. Taking for y an element of order n mod G' , and using c = 0 and g arbitrary, we see that (iii) holds.

(iii)~(iv): Since both (iii) and (iv) are expressible by sets of universal first- order formulas, it is clear that it will suffice to prove that (iii) ~ (iv) in the case when G is finitely generated. In this case we can choose X l , . . . , x~ ~ G such that {~1 . . . . . Y~} is a basis for (~.

Suppose now that we have

y~', - l~I g~,,(mod G') for 1 -- j - m and r~c~ i = niq~ for all i, j, (7) i = l

52

and we want to show that

D. SARACINO ALGEBRA UNIV.

l?I [y~, g~] = 1. (8) i=1

We can write

Yi- lZI x~ k~ and g i - Is/ xfk ~' (bothmod G'). (9) k = i k = l

If we modify these products by deleting all xk's that have finite order mod G', we obtain new yi's and g~'s for which (7) still holds; and by assumption (iii), the modification does not affect the truth of (8), since the deleted factors were central. We can therefore assume that all the xk's involved in (9) have infinite order mod G'.

Substituting the expressions for yj and g~ into (7) yields

Ylxp~,~,-I-I(x~&~.) for each j.l<-j<-m, so nieki=Xifk~ci~ (10) k k

since each 2k has infinite order. On the other hand

so we will be done if we can show that all the exponents in this last expression are 0. But by (10),

Nj ( e j , ~ - e J u i ) = 2~. 1 ( fvi2f i f~icij - f~jXdv~c~i ) J ni

=.,~i~q ~. (fvifu, c,i)-- ~,j.,~i ~. (fv'uic,i) = O,

as we see by using (7) to replace cij/nj by ql/ni in the first term. (iv) ~ (i): This is a compactness argument. Let G be any nil-2 group for which

(iv) holds, and let L be the first-order language for group theory augmented by constant symbols { g l g ~ G} and {g~ lg~G,~r a finite sequence of positive


integers}. Le t K be the L- theory

K = T2 U Diagram(G) U {g(~) = g I g ~ G}

U { ( g ~ , ) ~ = g,, [ g 6 G, r e 7 +, o- arbitrary},

where g(~ is g~ for the tr consisting of 1 alone, and o-'- 'r is o- with an r tacked on. K is consistent, for if K0 is any finite subset of K, Ko mentions only finitely many members of G, say gl . . . . , gin- If n is the product of all the integers occurring in all the sequences ~- mentioned in Ko, and n is the m-tuple n = (n, n . . . . , n), then since G ~ m . . ( g ) , Theorem 2.1 tells us that there is a nil-2 extension H of G containing nth roots ~gi for gj, 1 - - j - - m . If for each gi mentioned in Ko we interpret gj. in H as (~/gi) ~/~('~, where ~-(~-) is the product of the integers in ~- (and interpret gj as gj), we obtain a model for Ko. Thus, by the Compactness Theorem, K has a mode l /3 .

If C is the subgroup of /3 generated by the interpretations of all the g~'s, then C is a divisible extension of G. Divisibility is proved by induction on the number of factors g~ and (g~)-i occurring in an expression for an element of C. Clearly (the interpretation of) any g~ or (g~)-x has a kth root for any k > 0. Assuming that w E C and w has a 2k th root v, then for any g~ (and similarly for (g~)-~) we note that by 1.1(ii)

Wgcr = (1)gcr~(2k))2k['/), gcr~(2k)] k(2k-1),

the kth power of an element of C. [ ]

Theorem 2.3 can be viewed as a generalization of the case c ---2 of a theorem of Malcev [4], which states that every torsion-free nilpotent group of class c can be embedded in a divisible group of class c. (It is easy to see that every torsion-free nil-2 group satisfies condition (iii) of 2.3.)

The fact that (iii) ~ (i) in 2.3 can be proved without the detour through (iv): One reduces the problem to showing that if G is a finitely generated nil-2 group satisfying (iii) and gl . . . . ,gm s G are such that {gl . . . . , ~,,} is a basis for G, then for any n > 0 there is a nil-2 extension H of G which contains an nth root for each gi (this is essentially done by using the compactness argument given above). One then modifies the proof of 2.1 to suit such G and g, by replacing the bik's by products 1-I g~'~ and arguing that go = 1 by writing go in terms of the commutators [gl, gi], i < j . (One uses (iii) to handle those [g~, gi] for which either g~ or gi has finite order).

Yet another proof that ( i i i )~ (i) can be obtained by showing that if G is a nil-2 group satisfying (iii) and g ~ G, then for any n > 0 there is a nil-2 extension


H of G which contains an nth root for g and satisfies (iii). However, we have only been able to carry this through by reducing to the finitely generated case and then successively adjoining roots for elements gl, .. �9 g,,, chosen as in the preceding paragraph.

3. Amalgamation bases

Again let Tz denote the theory of nil-2 groups. A nil-2 group G is called an amalgamation base for Tz if for every two nil-2 extensions A and B of G there exist a nil-2 group H and embeddings qJA: A--> H and qJB:B--> H such that t)A(g) = tkB(g) for all g e G. G is called a strong amalgamation base if H, tOA, and ~ can always be chosen so that in addition ~A(A) Cl OB(B) = tOA(G). Clearly G is a strong amalgamation base iff for every two nil-2 extensions A, B of G such that A VI B = G, there is a nil-2 group H containing both A and B as subgroups.

Our main goals in this section are to characterize the amalgamation bases for Tz, to show that every amalgamation base for T2 is strong, and to determine in which cardinalities amalgamation bases for T2 exist. We have organized our results with the application to [5] in mind.

L E M M A 3.1. Suppose G is a nil-2 group and A, B are nil-2 extensions of G such that A Cl B = G. Suppose that

(i) A ' N G = G ' and B ' N G = G '

(ii) GA' /A ' and GB'/B' are summands of .~ and B, respectively, so that we have

= GA' /A ' ~ U/A' and /~ = GB'/B' ~ V/B'

for some subgroups U c_ A, V c_ B such that A ' c_ U and B' c_ V. Then there exists a nil-2 group H containing both A and B as subgroups.

Proof. We shall work in A (2) B, and to avoid confusion it will be convenient to suppose that instead of G ~_ A and G ~ B we have G c A and an isomorphism ~p from G onto a subgroup r of B.

In A (~)B, let N be the smallest normal subgroup containing {gq~(g-1) [ g s G}. To prove the lemma, it will suffice to show

If a s A , b~B, and ab~N, (11)

then

a s G and b=q~(a-1).

To verify (11), suppose that

I-I ((a~b~)(g~hyl)~'(a~b~) -~) = ab,


where a~ e A, bi ~ B, e~ = +1, and h~ -- r Writing a~ = c~u~, b~ = ~v~, with c~ e G, u~ e U, d~ ~ ~o(G), and v~ e V, this becomes

E (E(cluidivi) -1, (gihT, 1)-~'](glh~l)~') = ab.

If we write the left side in the form a/3y and use uniqueness, we obtain

a = 1-I [c~, g~,] lq [~, g~,] 17I g~,, b = 1-I [a,., h~-~,] l-I [,~, h; ~'3 l-I h;',, (12)

and

(61-I [c,, hi -~,] I-I [g;-% d~])(II [g~-% v,])(I-I [u~, h;-',]) = 1, (13)

where 8 = (1-I (g~h;-1)~')(rI h?~')-~(I-[ g~0 -~ is in [G, re(G)] since 6 can be written as a product of commutators [g~, hi] and their inverses.

Now by assumption (i), GA'/A ' -~ G via the correspondence g A ' - * gG', and likewise q~(G)B'/B '= q;(G). Thus we can write assumption (ii) as

f i ~ = G ( ~ U * and B = q ~ ( G ) ~ V * ,

so by a standard isomorphism,

fi~ | - (G | q~(G)) ~ (G | V*) (~ (U* | q~(G)) (9 (U* | V*). (14)

We also have homomorphisms

| q (G) ~ G ' such that

| V* - . B ' such that

U* | r --~ A ' such that

gl | ~ [gl, g~J, g | ~--~[~o(g), v], and

a | ~ ( g ) ~ [ u , g],

(15)

(1-[ (gig;-1)~,(r[ g;-~,)-l(I] g~)-l)(l-[ [c~, g~-~,])(II [g;-% ~-~(~)]) = 1,

1 - I [h7%v~]= l , and l [ I [u~,g; -~]=l (16)

where fi, 73 are the images of u, v in A, B respectively. For instance, the second of these maps is induced by the bilinear map G x V*---~B' given by (g, ~5)---~ [,p(g), v].

If we take Eq. (13) and apply first MacHenry 's isomorphism, then (14), and then (15) we obtain, corresponding to the three factors on the left side of (13)


Taking inverses in the last two equations and using (12) yields

a = I--[ [c,, g~,] YI g~, e G and b = I I [d,, hi -~,] I-[ h7%

and to finish it suffices to show that ar -- 1, i.e.,

1-I [c,, gZ,] 1-I 1-I [r 1-I gO" = 1.

But we can obtain this from the first of Eqs. (16) by cancelling the term 1-I (g;g?-l)~, �9 and taking inverses on both sides. []

Remarks. After we proved 3.1 we learned that the proof could be shortened by applying a result of J. Wiegold [7, Theorem 8.13]. Wiegold's result gives necessary and sufficient conditions for the truth of (11) in terms of the existence of homomorphisms with certain properties, defined on subgroups of A | B (which is isomorphic to fi, | Wiegold's proof is given for arbitrary, not necessarily nil-2 groups A and B (in this case A Q ) B is F/[F, [A, B]], with F = A * B). Since his proof is fairly lengthy, we have retained the above direct proof. The direct proof also facilitates the following extension of 3.1, which will be applied in [5].

L E M M A 3.2. Suppose that in condition (ii) of Lemma 3.1 we add the assumption that U/A' and V/B' are direct sums of cyclic groups:

U/A'= (~ (~) and WB '= ~ (~i)- i i

Then the group H in the conclusion of 3.1 can be chosen to have the property that if

ab YI [u~, v~] k~, = 1, (i j )

where a ~ A, b e B, and the product is taken over distinct pairs (i, j), then for each (i, j), ~i is divisible by the g.c.d, of the orders of ~ and ~j. (We adopt the convention that g.c.d.(m, oo) = m for m > 0 and g.c.d.(% oo) = 0.)

Proof. The group H constructed in the proof of 3.1 has the desired property. For if ab I-I [u~,v~] k'i = 1 in H then in A ( ~ B we have

abYI [u~, vi]k,, ~ N. (17)

We saw in the proof of 3.1 that if we apply MacHenry's isomorphism and then


(14) to the [A, B]-factor of an element of N (represented by the left side of (13)), we get no U* | V* term. Thus (17) implies that ~ (/q-i~- | ~3i) = 0 in U* | V*, and since U* | V*-~Et~(i,i~ ((~} ~ (vi}) the lemma follows. []

T H E O R E M 3.3. Let G be a nil-2 group. The following are equivalent: (i) G is an amalgamation base for T2

(ii) G satisfies (I) G' = Z(G), and (II) Vg ~ G Vn > 0 (g ~ G " G ' v 3 y ~ G 3 k ~7/ (y" - gk(mod G')/x[y, g]7 ~ 1))

(iii) G is a strong amalgamation base for T2.

Proof. We must show that (i) ~ (ii) ~ (iii). (i) ~ (ii): Suppose (I) fails, so that there is some g s Z ( G ) - G'. Let D be any

nil-2 group containing a commutator d which has the same order as g. If M is the normal subgroup of G x D given by M={(gm, d - ' ) [ meT/}, then ( G x D ) / M provides us with a nil-2 extension A of G in which g is a commutator. On the

o t h e r hand, if (x) is an infinite cyclic group, then B = G ( ~ ( x ) gives us an extension of G in which [g, x] 7 ~ 1, since ~ | ~ ~ 0 in G | (~) because gr G' . Thus G is not an amalgamation base, because if tO A, qs s were embeddings of A, B into a nil-2 group H which agreed on G we would have [0B(g), rOB(x)] ~ 1 in H although OA(g) e H ' , a contradiction.

Suppose (II) fails for some g ~ G and n > 0 . Then for all k, G ~ V y (y~=-gk(mod G ' ) ~ [y, g] = 1), so by Corollary 2.2 there is a nil-2 extension A of G containing an nth root ~g for g. On the other hand the fact that g~ G"G' tells us that if (x) is a cyclic group of order n then g|162 0 in G@(~), so in B = G(~(x) we have [g, x ] ~ 1 and x" = 1. If t)A, t)B were embeddings of A, B into some nil-2 H which agreed on G we would have in H

1 ~ [4~B (g), rOB (x)] = [(~A (#g))~, CB (x)] = [0A (#g), CB (x")] = [~A (#g), 1] = 1,

a contradiction. Thus G is not an amalgamation base. (ii) ~ (iii): Suppose G satisfies conditions (I) and (II), and let A, B be any two

nil-2 extensions of G such that A fq B = G. We claim that there is a nil-2 group H containing A a n d / 3 as subgroups.

We can assume that A and B are both finitely generated over G, since by a standard compactness argument on diagrams it will suffice to handle this case.

We now apply Lemma 3.1. Condition (i) of 3.1 is satisfied since it is clear that A' f"l G c Z(G), so A ' f ' /G = G' by (I), and likewise B' Cl G = G'. To verify condition (ii) of 3.1 we first note that GA' /A ' is a pure subgroup of A. For if g~G, a ~ A . and a ~ = g ( m o d A ' ) , then for any y ~ G and k~7/ such that


y"- -gk(mod G') we have in A

Ey, g] = [y, a" ] = Ey ", a] = a] = Ea a] = 1.

Thus by (II) we must have g ~ G n G ', so there is some g l ~ G such that g~-- g (rood A').

Since G A ' / A ' is pure in ,S,, we can conclude that it is a summand once we know that the quotient of fi~ by GA' /A ' has a basis ([2], Theorem 5). But the quotient does have a basis, since it is finitely generated (becacause A is finitely generated over G).

The same argument shows that GB'/B' is a summand of/~. []

In some special cases, conditions (I) and (II) for amalgamation bases can be reduced to just (I):

PROPOSITION 3.4. Let G be a nil-2 group of exponent 4, or of exponent p for some prime p. Then G is an amalgamation base for T2 iff G' = Z(G) .

Proof. We give the proof for exponent 4, the case of exponent p being similar but easier. Clearly we must show that if G ' = Z(G) and G has exponent 4 then

(II) holds for G. By Proposition 1.1(ii) we have the identity (xy)4=x4y4[y,x] 6 in any nil-2

group G. If G has exponent 4 this becomes 1 = [y, xZ], so all squares in G are central. By the assumption G' = Z(G) this means that G 2_ G'.

Now take any g s G and any n > 0 . If 2~ 'n then g ~ G ", since n is then relatively prime to the order of g. If g ~ Z(G) then g ~ G ' by assumption. So if 2 X n or g ~ Z ( G ) then g e G"G '. In the remaining case where 2 I n and g d Z ( G ) we can choose a y ~ G such that [ y , g ] ~ l ; since 2 I n and G2~_G ' we have y~ -- g~ G'), so we see that (II) holds by using k = 0. []

EXAMPLE. Condition (I) does not imply condition (II) for groups of exponent 8 or p2 (p>2) . In fact, for any p, let (x),(y}, and (z) be cyclic groups of orders p2, p2, and p respectively, and let G = (x)(~)(y)(~)(z). Every element of G has a unique expression in the form

x"ybzC[x,y]a[x,z]'[y,z~, with O<--a,b,d<p 2 and O<-c,e, f<p. (18)

It follows that (1) G has exponent 8 if p = 2 and exponent p2 if p > 2. (2) G ' = Z ( G ) = t h e set of elements in whose representation (18) we have

a = b = c = O .


(3) zr GPG ' and Vw ~ G Vk(w p = zk(mod G') ~ [w, z] = 1), so (II) fails for

G.

EXAMPLES. Let p be a prime. Then any group of order p or pZ is nontrivial abelian, hence violates condition (I) and is not an amalgamation base for Tz. On the other hand, every nonabelian group of order p3 is an amalgamation base.

First of all, it is easy to see that any such group G is nil-2 and has Z(G) and G ' both of order p, so condition (I) holds. If p = 2 the only possibilities for G are the group of unit quaternions and the dihedral group of order 8, each of which has exponent 4 and is thus an amalgamation base by 3.4. If p > 2 there is (see [1], p. 52) one possibility for G of exponent p, which is covered by 3.4, and one possibility for G of exponent p2. This last group G is generated by elements a and b subject to the relations a p~ = 1, b p = 1, and [a, b] = aL Every element g ~ G can be written in the form g = a~b i, where 0--< i < p2 and 0-<] < p.

To verify (II) for G, let n > 0. If p % n or if p divides both i and ] in g = a~b i then g s G'~G '. If p I n and p k" i or p Xj then g fails to commute with either b or a (since [a, b] has order p), and since we have a" - b" = g~ G'), (II) holds.

It is possible to extend Proposition 3.4 slightly by using the following result, which also enables us to determine in what finite cardinalities amalgamation bases for T2 exist. (Of course, amalgamation bases exist in all infinite cardinalities, because existentially complete models are amalgamation bases.)

T H E O R E M 3.5. Let G, H be nil-2 groups of relatively prime exponents c and d, respectively. Then G ~ H is an amalgamation base for Tz iff both G and H are.

Proof. It is easy to see that in general, if G �9 H is an amalgamation base then so are G and H.

Conversely, suppose G, H are amalgamation bases of relatively prime exponents c, d. It is clear that since condition (I) of Theorem 3.3 then holds for both G and H, it holds for G ~ ) H as well. We now check condition (II).

Let (g, h)~ G ~ H and let n > 0 . By (II) for G and H we have for G: either (1) a" = g(mod G') for some a ~ G or

(2) 3x ~ G 3 kl(x" =- gk'(mod G')/x Ix, g] ~k 1) and likewise for H: either (3) b" = h(mod H') for some b ~ H or

(4) 3y ~ H 3k2(y" - h%(mod H' )A[y , h] ~ 1). We check that (II) holds for (g, h) and n by considering cases. If (1) and (3) hold then (a, b) n --- (g, h)(mod (G �9 H)') , and we are done.

60 D SARACINO ALGEBRA UNIV.

If (1) and (4) hold (and similarly if (2) and (3) hold) then in G ~ H

(a ~, y)" ----(g, h)~(mod ( G ~ H ) ' ) and [(a ~, y), (g, h ) ]~ 1,

and we are done. If (2) and (4) hold we have

x" -- gk,(mod G') A[x, g] ~ 1 and y~ - h~(mod H') A[y, h] ~ 1. (19)

We can assume that ka and k 2 a r e relatively prime to d and c, respectively. For instance, since c and d are relatively prime we can write kl = k3m, where k3 is relatively prime to d and m is relatively prime to c. Then, choosing an integer u such that mu--1(rood c) we obtain (xU) " - gk3(mod G'); and [x ~', g ] ~ 1, because [x, g ] " = 1 together with [x, g]C= 1 would yield [x, g] = 1 (since u is relatively prime to c). Thus if kl is not relatively prime to d at the outset we can replace x and kl by x ~ and k3. A similar argument allows us to assume that k2 is relatively prime to c.

Now from (19) we get

(x%)" ~gk,k2(mod G') and (yk,), = hk,%(modH,);

thus (x k~, yk,),, _ (g, h)kl~(mod (G ~ H)') and to finish it suffices to show that we do not have

[(xk2, yk,), (g, h)] = 1 in G ~ H. (20)

But ff (20) held we would have [x, g]~ = 1 in G, which together with [x, g]C = 1 would yield Ix, g] = 1 (since kz is relatively prime to c), a contradiction. []

T H E O R E M 3.6. Let G be a nil-2 group of exponent n, where n is either a

product of distinct primes or twice such a product. Then G is an amalgamation base

for WE iff G' = Z (G) .

Proof. We must show that if G ' = Z ( G ) then G is an amalgamation base. Since G is the direct sum of its maximal p-subgroups (e.g., by [6], Theorem VIA.w) we can write

G = G I ~ " " "~Gr ,

where the exponents of the Gi's are 4 or prime, and pairwise relatively prime.


Since G ' = Z ( G ) , G ' = Z ( G I ) for each i, so each G~ is an amalgamation base by Proposition 3.4. Thus G is an amalgamation base by Theorem 3.5 and

induction. []

We now consider the problem of finding amalgamation bases of various finite

cardinalities.

L E M M A 3.7. Let p be a prime and k a positive integer. I f k = 1, 2, or 4 there are no amalgamation bases for Ta of cardinality pk. I f k ~ 1, 2, or 4 then there exists an amalgamation base which has cardinality pk and which has exponent p (for

p > 2) and exponent 4 (for p = 2).

Proof. We have already seen that there are no amalgamation bases of cardinality p or p2. A case-by-case inspection of the nil-2 groups of cardinality p4 (see

for instance [6], Table 1, p. 197) shows that they all violate condition (I), hence are not amalgamation bases.

Now suppose k ~ 1, 2, or 4. If k is odd then k -- 3, so we can write k = 2m + 1, with m -> 1. Let G be the direct sum of m copies of the cyclic group 7/0 of order p. Then the group G~)Z~ has order p2m§ = pk; its exponent is p, if p > 2 , and 4, if p - - 2 . This group also satisfies G ' = Z ( G ) , so it is an amalgamation base by Proposition 3.4.

Now suppose k is even. Since k-> 6 we can write k = ka + k2 where both kl and k2 are odd and -----3. By the preceding paragraph we can let G1 and G2 be amalgamation bases of cardinalities pk, and p~ and exponent p or 4, depending on p. Then by Proposition 3.4 we see that G1 �9 G2 is an amalgamation base of cardinality pk and appropriate exponent. [ ]

It is now easy to determine for which positive n's there exist amalgamation bases of cardinality n. Since the trivial group is clearly an amalgamation base, we restrict oursleves to n > 1.

T H E O R E M 3.8. Let n > 1 and let n = p~ . . . . p~, be the prime-power decom- position of n. Then there exist amalgamation bases for T2 of cardinality n iff no lq is 1, 2, or 4. I f this condition is satisfied, then there exists an amalgamation base which has cardinality n and which has exponent Pl " " " Pr (for odd n) and exponent

2pl "" �9 Pr (for even n).

Proof. This follows from 3.5 and 3.7. We omit the details. [ ]

62 O. SARACINO ALOEBRA UNIV.

�9 Postscript. I t has c o m e to ou r a t t en t ion tha t the e q u i v a l e n c e b e t w e e n i t ems (i)

and (iii) in T h e o r e m 2.3 was p rev ious ly p r o v e d by Pau l C o n r a d in " C o m p l e t i o n s

of g roups of class 2" , I l l inois J. Ma th . 5 (1961), 2 1 2 - 2 2 4

REFERENCES

[1] M. HALL, The Theory of Groups, Macmillan, New York, 1959. [2] I. KAr'LA~SKY, Infinite Abelian Groups, University of Michigan Press, Ann Arbor, 1969. [3] T. MACHENRu The tensor product and the 2nd nilpotent product of groups, Math. Z. 73 (1960),

134-145. [4] A. I. MAL'CEV, Nilpotent torsion-free groups, Izv. Akad. Nauk SSSR Set. Mat. I3 (1949), 201-212. [5] D. SARACrNO and C. WOOD, QE nil-2 groups of exponent 4, to appear in Journal of Algebra. [6] E. SCHENKMAN, Group Theory, Van Nostrand, Princeton, NJ, 1965. [7] J. WrEGOLD, Nilpotent products of groups with amalgamations, Publ. Math. Debrecen 6 (1959),

131-168.

Colgate University Hamilton, New York U.S.A.

amalgamation bases for nil-2 groups

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