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On Product-Equality-Preserving Mappings in Groups
JOURNAL OF ALGEBRA ARTICLE NO.
182, 653]663 Ž1996.
0194
On Product-Equality-Preserving Mappings in Groups Gadi Moran Department of Mathematics and Computer Science, The Uni¨ ersity of Haifa, Haifa 31905, Israel Communicated by Leonard Lipshitz Received March 13, 1995
Let G, H be groups, M : G. A mapping f : M ª H is called a Product-equalitypreser¨ ing ŽPEP. mapping iff it satisfies ;a, b, c, d g M w ab s cd « f Ž a . f Ž b . s f Ž c . f Ž d . x .
Ž). M
THEOREM. Let G1 , G 2 be nonabelian groups with centers Z1 , Z2 and let M1 , M2 satisfy Gi _ Zi : Mi : Gi ,
i s 1, 2.
Let f : M1 ª M2 be a PEP mapping that maps M1 onto M2 . Then there is an epimorphism w : G1 ª G 2 and ¨ g Z2 such that for every x g M1 , f Ž x . s ¨w Ž x . . Q 1996 Academic Press, Inc.
0. INTRODUCTION 0.0. Parameters of the multiplication table of a subset M of a group G often make a useful research tool. Typically, the number < M 2 < of distinct entries of this table for finite M is such a parameter. On one hand, an extensive body of research deals with determination of the possible structure of M when < M 2 < is small in specific groups, e.g., the additive group of integers ŽwF1x, wF3x.. On the other hand, interesting classification theorems for groups G, where < M 2 < - < M < 2 for every M : G of a fixed finite cardinality m, are available. See wF2x for m s 2, wBFPx for m s 3, and wBx, wBFx, and wFSx for related topics and further references. In comparing multiplication tables of different group subsets M, M9, a bijection between M and M9 that, along with its inverse, preserves product equality Ži.e., a PEP bijection with PEP inverse. plays an important role. Such a mapping is called an isomorphism by Freiman in wF1x and a 653 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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GADI MORAN
2-isomorphism in wFSx. See wBx for further remarks and open questions concerning this notion. A convenient setting for a systematic treatment of this line of research is provided by a category that we shall denote PEP Žsee wJx, Vol. II, Chapter 1, for basics of category theory.. The objects of PEP are ordered pairs Ž M, G ., where M is a subset of the group G. A PEP morphism from Ž M, G . to Ž M9, G9. is any PEP mapping f : M ª M9. It will be convenient to use M alone to denote the object Ž M, G . whenever the context allows. Let us make some observations. We let G, H, K denote groups in the sequel, and use w , c to denote group homomorphisms. 0.1. If M1 : G, M2 : H, w : G ª H is a homomorphism, and w Ž M1 . : M2 , then f [ w < M1 , the restriction of w to M1 , is a PEP mapping from M1 to M2 . 0.2. For h g H, let h LŽ h R . denote left Žright. multiplication by h in H. That is, h L , h R : H ª H are defined by h LŽ y . s hy,
h R Ž y . s yh
Ž ygH..
Let h9, h0 g H. If M : H satisfies hXL < M s hYR < M,
Ž 0.1.
then f [ hXL < M s hYR < M is a PEP mapping. Indeed, if a, b, c, d g M and ab s cd, then f Ž a . f Ž b . s hXLŽ a . hYR Ž b . s h9abh0 s h9cdh0 s hXLŽ c . hYR Ž d . s f Ž c . f Ž d . . 0.3. Let M1 : G, M2 : H, h9, h0 g H, and let w : G ª H be a homomorphism. We call f : M1 ª M2 the standard Ž h9, w , h0 .-PEP mapping iff f s hXL ( w < M1 s hYR ( w < M1 .
Ž 0.2.
That is, f Ž x . s h9w Ž x . s w Ž x . h0 for all x g M1. By 0.1 and 0.2, f is indeed a PEP mapping whenever f is the standard Ž h9, w , h0 .-PEP mapping. We shall call f the standard Ž h, w .-PEP mapping if h s h9 s h0 and f is the Ž h9, w , h0 .-PEP mapping. Thus, f : M1 ª M2 is the standard Ž h, w .PEP mapping if f Ž x . s hw Ž x . s w Ž x . h
Ž x g M1 . .
Ž 0.3.
Notice that if f : M1 ª M2 is the standard Ž h9, w , h0 .-PEP mapping and for some x g M1 , h9w Ž x . s w Ž x . h9, then in fact h9 s h0 and f is actually the standard Ž h, w .-PEP mapping, where h s h9 s h0.
PRODUCT-EQUALITY-PRESERVING MAPPINGS
655
We say that f : M1 ª M2 is a standard PEP mapping if it is the standard Ž h9, w , h0 .-PEP mapping for some h9, w , h0. 0.4. The composition of standard PEP mappings is a standard-PEP mapping. Indeed, if f : M1 ª M2 is the standard Ž h9, w , h0 . mapping, M3 : K, and g: M2 ª M3 is the standard Ž k9, c , k0 . mapping, then gf : M1 ª M3 is the standard Ž k9c Ž h9., cw , c Ž h0 . k0 .-PEP mapping. Indeed, for x g M1 , gf Ž x . s g Ž h9w Ž x . . s k9c Ž h9w Ž x . . s Ž k9c Ž h9 . . cw Ž x . , gf Ž x . s g Ž w Ž x . h0 . s c Ž w Ž x . h0 . k0 s cw Ž x . Ž c Ž h0 . k0 . . Similarly, if f : M1 ª M2 is the standard Ž h, w . mapping, g: M2 ª M3 is the standard Ž k, c . mapping, and k c Ž h. s c Ž h. k, then gf is the standard Ž k c Ž h., cw .-PEP mapping. 0.5. If f : M1 ª M2 is the standard Ž h1 , w , h 2 .-PEP mapping, w : G ª H is a group isomorphism Žso that f is one to one., and f maps M1 onto M2 , then fy1 : M2 ª M1 is also a PEP mapping. In fact, it is the standard Ž wy1 Ž hy1 ., wy1 , wy1 Ž hy1 ..-PEP mapping, as one readily checks. 1 2 Similarly, if f : M1 ª M2 is the standard Ž h, w .-PEP mapping, w : G ª H is an isomorphism Žso that f is one to one., and f maps M1 onto M2 , then the bijection fy1 : M2 ª M1 is also a PEP mapping, and, in fact, it is the standard Ž wy1 Ž hy1 ., wy1 .-PEP mapping. 0.6. Let M : G satisfy ;a, b, c, d g M w ab s cd « a s c & b s d x .
Ž 0.4. M
Then any f : M ª H is a PEP mapping, as Ž).M holds trivially. It follows that the inverse of a PEP bijection is not, in general, a PEP mapping. Ž0.4.M holds, for instance, when M freely generates G or when M s Ž a, y .: y g Y 4 is a set of Ždistinct. transpositions, all moving the element a, and G is a group of permutations of a set containing a4 j Y, including M. In contrast with 0.6, we have the following two theorems. The first theorem was proved by Freiman wF3x. THEOREM 1. Let G, H be groups. Then for e¨ ery PEP mapping f on G into H there is a homomorphism w : G ª H and ¨ g H centralizing w Ž M . such that f Ž x . s ¨w Ž x .
Ž xgM..
656
GADI MORAN
Let us say that M : G is cocentral in G iff G _ ZŽ G . : M, where ZŽ G . [ ¨ g G: ; x g Gw ¨ x s x¨ x4 is the center of G. Our main result is: THEOREM 2. Let G1 , G 2 be nonabelian groups and let Mi : Gi be cocentral in Gi , i s 1, 2. Then for e¨ ery PEP mapping f of M1 onto M2 there is an epimorphism w : G1 ª G 2 and ¨ g ZŽ G 2 . such that f Ž x . s ¨w Ž x .
Ž x g M1 . .
Theorems 1 and 2 are proved in Sections 1 and 2, respectively. The argument for Theorem 1 in Section 1 is the one we used in June 1993, en route toward the proof of Theorem 2 and unaware of Freiman’s work wF3x. It is included here for the reader’s convenience, as we feel it is a useful introductory vehicle for the more delicate argument presented for Theorem 2 in Section 2. Corollaries 0.7. By Theorem 1, the group of PEP bijections of a group G is the subgroup of SG , the symmetric group on the set G, generated by the group AutŽ G . of automorphisms of the group G and the set ¨ L : ¨ g ZŽ G .4 of Žleft. translations by central elements of G. Thus it is a subgroup of the holomorph HolŽ G . generated in SG by AutŽ G . and the set of left translations g L : g g G4 Žsee wJx, Vol. I, 1.10.. In particular: 0.8. If G is abelian, then the group of PEP bijections of G is HolŽ G .. 0.9. If G is nonabelian with center Z, then for every cocentral M in G, the group of PEP bijections of M is isomorphic to the group of standard PEP bijections of G mapping M onto M. In fact f ¬ f < M is such an isomorphism, by Theorem 2.
2. PROOF OF THEOREM 1 We assume in this section that G, H are groups and f : G ª H. We call f a PEP mapping iff Ž). G is satisfied, i.e., ;a, b, c, d g G ab s cd « f Ž a. f Ž b . s f Ž c . f Ž d . . We let 1 denote the identity element of G.
Ž 1.0.
PRODUCT-EQUALITY-PRESERVING MAPPINGS
657
1.1. The following are equi¨ alent: Ži. Žii. Žiii.
f is a PEP mapping, i.e., Ž1.0. holds. f Ž abcy1 . s f Ž a. f Ž b . f Ž c .y1 for all a, b, c g G. f Ž ab. f Ž1. s f Ž a. f Ž b . for all a, b g G.
Proof. Ži. « Žii.: Assume Ž1.0. and let a, b, c g G. As Ž abcy1 . c s ab, we have f Ž abcy1 . f Ž c . s f Ž a. f Ž b . and so Žii. holds. Žii. « Žiii.: Assume Žii. and let a, b g G. By Žii., f Ž ab. s f Ž ab1y1 . s f Ž a. f Ž b . f Ž1.y1 and so Žiii. holds. Žiii. « Ži.: Assume Žiii. and let a, b, c, d g G satisfy ab s cd. Then by Žiii., f Ž a . f Ž b . s f Ž ab . f Ž 1 . s f Ž cd . f Ž 1 . s f Ž c . f Ž d . . Thus, Ži. holds. 1.2. i.e.,
Let f : G ª H be a PEP mapping. Then f Ž1. g H centralizes f Ž G ., f Ž g . f Ž 1. s f Ž 1. f Ž g .
for all g g G.
Ž 1.1.
Proof. Indeed, by 1.1Žiii., f Ž g . f Ž 1. s f Ž 1 ? g . f Ž 1. s f Ž 1. ? f Ž g . . 1.3.
Let f : G ª H be a PEP mapping and define w : G ª H by
w Ž g . [ f Ž 1.
y1
fŽ g..
Then w Ž xy . s w Ž x . w Ž y . for all x, y g G, i.e., w is a homomorphism. Proof. By 1.1 and 1.2, we have
w Ž xy . s f Ž 1 .
y1
f Ž xy . s f Ž 1 .
y2
s f Ž 1.
y2
f Ž x . f Ž y . s Ž f Ž 1.
f Ž 1 . f Ž xy . s f Ž 1 . y1
f Ž x . .Ž f Ž 1 .
y2
y1
f Ž xy . f Ž 1 .
f Ž y. . s w Ž x. w Ž y. .
Proof of Theorem 1. Let f : G ª H be a PEP mapping. Define w : G ª H, ¨ g H, by
w Ž x . s f Ž 1.
y1
f Ž x. ,
¨ s f Ž 1. .
Then f Ž x . s ¨ w Ž x . and by, 1.2 and 1.3, w is a homomorphism and ¨ centralizes w Ž G .. Theorem 1 is proved.
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GADI MORAN
2. PROOF OF THEOREM 2 We assume in this section that G1 , G 2 are nonabelian groups with centers Z1 , Z2 , Mi is a cocentral subset of Gi , i.e., Gi _ Zi : Mi : Gi , i s 1, 2, and f : M1 ª M2 is a PEP mapping of M1 onto M2 , i.e., ;a, b, c, d g M1 ab s cd « f Ž a. f Ž b . s f Ž c . f Ž d . .
Ž 2.0.
Under these assumptions, we prove: THEOREM 2.0. There is an epimorphism w : G1 ª G 2 of G1 onto G 2 and an element ¨ g Z2 such that ; x g M1 f Ž x . s ¨ w Ž x . .
Ž 2.1.
Let us put Mi0 s Gi _ Zi , so that Mi0 : Mi , i s 1, 2. We proceed through a series of propositions. The first lists useful properties of the sets Mi0 . 2.1. Ži. Žii. Žiii. Živ.
Mi0 satisfies for i s 1, 2: a g Mi0 m ay1 g Mi0 . c g Mi0 « 'a, b g Mi0 w c s ab x. w g Zi , a g Gi « w a g Mi0 m aw g Mi0 x. Let a, b g Gi . Then ab g Zi « ab s ba.
Proof. Recall that a g Mi0 if ' x g Gw xa / ax x. Hence; Ži. Follows from xa / ax m ay1 x s ay1 Ž xa. ay1 / ay1 Ž ax . ay1 s y1 xa . Žii. Let b, c g Gi , cb / bc. Thus b, c g Mi0 and c s Ž cby1 . b. Let a s cby1 . Then c s ab and b g Mi0 . Also, a g Mi0 as by cb / bc we have ba s bcby1 / cbby1 s c s cby1 b s ab. Žiii. Indeed, Mi0 is a union of the cosets aZi in Gi other than Zi . Živ. Let a, b g Gi . As bŽ ab. by1 s ba, we see that ab s ba whenever ab commutes with b. Thus, if ab g Zi certainly ab s ba. 2.2. Ži. M20 : f Ž M10 .. Žii. There is a u g Z2 such that a, ay1 g M1 « f Ž a . f Ž ay1 . s f Ž ay1 . f Ž a . s u.
Ž 2.2.
Proof. Ži. Let c g M20 : M2 and let d g M2 satisfy cd / dc. As f maps M1 onto M2 , there are a, b g M1 such that f Ž a. s c and f Ž b . s d. By Ž2.0., ab / ba Želse cd s dc . so that a, b f Z1 , i.e., a, b g M10 . Hence c s f Ž a. g f Ž M10 . and we have M20 : f Ž M10 ..
PRODUCT-EQUALITY-PRESERVING MAPPINGS
Žii.
659
0 M10 / f , so let a0 g M10 . By 2.1Ži., ay1 0 g M1 as well. Put
u [ f Ž a0 . f Ž ay1 0 .. Let now a, ay1 g M1. As aay1 s ay1 a s a0 ay1 0 , we obtain, by Ž2.0., f Ž a . f Ž ay1 . s f Ž ay1 . f Ž a . s u and so Ž2.2. holds. We show that u g Z2 . Let a g M10 . By Ž2.2., we have f Ž ay1 . s uf Ž a.y1 s f Ž a.y1 u, so u commutes with f Ž a.y1 for all a g M10 . By 2.1Ži., u commutes with f Ž a. for every a g M10 , i.e., u centralizes f Ž M10 .. But M20 : f Ž M10 . by Ži., and so u centralizes M20 s G 2 _ Z2 , whence u g Z2 . 2.3. Ži. Žii. Žiii.
Let a, b, c g M1. We ha¨ e: ay1 g M1 « f Ž ay1 .y1 s uy1 f Ž a. s f Ž a. uy1 . abcy1 g M1 « f Ž abcy1 . s f Ž a. f Ž b . f Ž c .y1 . abc, c, cy1 g M1 « f Ž abc. s uy1 f Ž a. f Ž b . f Ž c ..
Proof. Ži. By 2.2Ži. f Ž a. f Ž ay1 . s f Ž a. f Ž ay1 . s u g Z2 , whence f Ž ay1 .
y1
s uy1 f Ž a . s f Ž a . uy1
Žii. By Ž2.0. and Ž abcy1 . c s ab, we have f Ž abcy1 . f Ž c . s f Ž a . f Ž b .
or
f Ž abcy1 . s f Ž a . f Ž b . f Ž c .
y1
.
Žiii. By Žii. and Ži., we have f Ž abc . s f ab Ž cy1 .
ž
y1
y1 y1
/ s f Ž a. f Ž b . f Ž c
.
s f Ž a . f Ž b . f Ž c . uy1 s uy1 f Ž a . f Ž b . f Ž c . . 2.4. There is a ¨ g Z2 such that a, b, ab, Ž ab .
y1
g M1 « ¨ s f Ž a . f Ž b . f Ž ab .
y1
s f Ž ab .
y1
f Ž a. f Ž b . .
Ž 2.3.
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GADI MORAN
We split the proof of 2.4 into steps: 2.4.1.
Let a, b, ab, Ž ab.y1 , a, ˜ ˜b, ab, ˜˜ Ž ab ˜˜.y1 g M1. Then f Ž a. f Ž b . f Ž ab ˜˜. s f Ž ab . f Ž a˜. f Ž ˜b . .
Ž 2.4.
Proof. Indeed, let c s abab ˜˜ s abŽ ab ˜˜. s Ž ab. ab. ˜˜ By 2.3Žiii., f Ž c . s uy1 f Ž a . f Ž b . f Ž ab ˜˜. s uy1 f Ž ab . f Ž a˜. f Ž ˜b . . Ž2.4. follows. 2.4.2.
Let a, b, ab, Ž ab.y1 g M1. Then f Ž a . f Ž b . f Ž ab .
y1
s f Ž ab .
y1
f Ž a. f Ž b . .
Proof. Put a s a ˜ and b s ˜b in 2.4.1. 2.4.3.
Let a, b, ab, Ž ab.y1 , a, ˜ ˜b, ab, ˜˜ Ž ab ˜˜.y1 g M1. Then f Ž a . f Ž b . f Ž ab .
y1
s fŽa ˜. f Ž ˜b . f Ž ab ˜˜.
y1
.
Ž 2.5.
Proof. By 2.4.1, Ž2.4. holds. Hence f Ž ab .
y1
f Ž a. f Ž b . s f Ž a ˜. f Ž ˜b . f Ž ab ˜˜.
y1
.
By 2.4.2, Ž2.5. holds. 2.4.4.
Let a0 , b 0 , a0 b 0 g M10 and let ¨ [ f Ž a0 . f Ž b 0 . f Ž a0 b 0 .
y1
.
Then ¨ g Z2 . Proof. Let d g M20 . By 2.2Ži., there is a c g M10 with f Ž c . s d and, by 2.1Žii., there are a, b g M10 with c s ab. By 2.1Ži., cy1 s Ž ab.y1 g M10 and also Ž a0 b 0 .y1 g M10 . Thus, by 2.4.3 and 2.4.2 ¨ s f Ž a0 . f Ž b 0 . f Ž a0 b 0 .
y1
s f Ž a . f Ž b . f Ž ab .
y1
s f Ž ab .
y1
f Ž a. f Ž b . ,
i.e., ¨ s f Ž a. f Ž b . dy1 s dy1 f Ž a. f Ž b ., whence f Ž a. f Ž b . s d¨ s ¨ d. Thus, ¨ centralizes M20 , and so ¨ g Z2 . Proof of 2.4. As M10 / f , 2.2Ži. and Žii. guarantee the existence of a0 , b 0 g M10 such that also a0 b 0 g M10 , hence also Ž a0 b 0 .y1 g M10 . Use
PRODUCT-EQUALITY-PRESERVING MAPPINGS
661
a0 , b 0 to define ¨ g G 2 as in 2.4.4. By 2.4.1]2.4.4, ¨ g Z2 , and whenever a, b, ab, Ž ab.y1 g M2 , we have ¨ s f Ž a . f Ž b . f Ž ab .
2.5. Ži. Žii. Žiii.
y1
s f Ž ab .
y1
f Ž a. f Ž b . .
Let w g Z1 , aw, a, ay1 , bw, b, by1 g M1. Then: f Ž aw . f Ž a.y1 s f Ž bw . f Ž b .y1 . f Ž aw . f Ž a.y1 g Z2 . f Ž aw . f Ž a.y1 s f Ž a.y1 f Ž aw ..
Proof. Ži. As w s Ž aw . ay1 s Ž bw . by1 , Ž2.0. implies f Ž aw . f Ž ay1 . s Ž f bw . f Ž by1 .. By Ž2.2., f Ž ay1 . s f Ž a.y1 u and f Ž by1 . s f Ž b .y1 u, so that f Ž aw . f Ž a.y1 s f Ž bw . f Ž b .y1 . Žii. Let d g M20 . By 2.2Ži., let c g M10 satisfy d s f Ž c .. As cw, c, cy1 g M10 : M1 , we have, by Ži., f Ž aw . f Ž a.y1 s f Ž cw . f Ž c .y 1 . Thus df Ž aw . f Ž a.y1 s df Ž cw . f Ž c .y1 s f Ž c . f Ž cw . f Ž c .y1 . Hence, by 2.3Žii. and by w g Z1 , df Ž aw . f Ž a .
y1
s f Ž c ? cw ? cy1 . s f Ž cw . s f Ž wc . s f Ž Ž aw . ay1 c . s uy1 f Ž aw . f Ž ay1 . f Ž c . s uy1 f Ž aw . f Ž a .
y1
ud s f Ž aw . f Ž a .
y1
d,
where the last three equalities use 2.3Žiii., and 2.2Ži.. Thus, f Ž aw . f Ž a.y1 centralizes M20 , hence f Ž aw . f Ž a.y1 g Z2 . Žiii. f Ž aw . f Ž a.y1 s f Ž a.y1 f Ž aw . follows from Žii. by 2.1Živ.. We are now ready to define the epimorphism w : G1 ª G 2 and the element ¨ g Z2 mentioned in Theorem 2.0. DEFINITION 2.6. Let a0 , b 0 , a0 b 0 g M10 . Define an element ¨ g G 2 and a mapping w : G1 ª G 2 as follows: Ži. Žii. Žiii.
¨ [ f Ž a0 . f Ž b 0 . f Ž a0 b 0 .y1 .
w Ž a. [ ¨ y1 f Ž a. for a g M10 . w Ž w . [ f Ž a0 w . f Ž a0 .y1 for w g Z1.
Two remarks are in order: 2.6.1. ¨ is well defined, ¨ g Z2 , and satisfies Ž2.3.. Proof. M10 / f , so let c g M10 . By 2.1Žii., there are a0 , b 0 g M10 with a0 b 0 s c, and by 2.1Ži., also Ž a0 b 0 .y1 s cy1 g M10 . M10 : M1 , so ¨ s f Ž a0 . f Ž b 0 . f Ž a0 b 0 .y1 is well defined. By 2.4, 2.6.1 follows.
662
GADI MORAN
2.6.2. Ži. a g M10 « w Ž a. s ¨ y1 f Ž a. s f Ž a. ¨ y1 . Žii. w g M1 _ M10 , a g M10 « w Ž w . s f Ž aw . f Ž a.y1 s f Ž wa. f Ž a.y1 s f Ž a.y1 f Ž aw . s f Ž a.y1 f Ž wa. g Z2 . Proof. By 2.6.1 and 2.5. ¨ of Definition 2.6 and u of 2.2 are related in the next proposition.
2.7.
u s ¨ 2.
Proof. Let a, b, ab g M10 . Then also ay1, by1 , Ž ab.y1 g M10 and, by 2.6.1, 2.2, and 2.3Ži.,
2.8.
¨ 2 s f Ž a . f Ž b . f Ž ab .
y1
s f Ž a . f Ž b . f Ž ab .
y1
f Ž Ž ab .
y1 y1
.
f Ž by1 . f Ž ay1 .
uy1 f Ž ab . uf Ž b .
y1
uf Ž a .
y1
s u.
f Ž x . s ¨ w Ž x . for x g M1.
Proof. If x s a g M10 , 2.8 holds by Definition 2.6Žii.. Let x s w g M1 _ M10 s M1 l Z1 and let a g M10 . Then w, a, wa, Ž wa.y1 g M1 , so by Ž2.3., ¨ s f Ž w . f Ž a. f Ž wa.y1 and by 2.6.2, w Ž w . s f Ž wa. f Ž a.y1 . Hence ¨ w Ž w . s f Ž w .. 2.9.
w : G1 ª G 2 is a homomorphism.
Proof. Let x 1 , x 2 g G1. We show w Ž x 1 x 2 . s w Ž x 1 . w Ž x 2 .. Distinguish the following possible cases: Case 1. we have
x 1 s a, x 2 s b, a, b, ab g M10 . By Definition 2.6Žii. and 2.6.1,
w Ž a . w Ž b . s ¨ y1 f Ž a. ¨ y1 f Ž b . s ¨ y1 Ž ¨ y1 f Ž a. f Ž b . . and ¨ y1 s f Ž ab. f Ž b .y1 f Ž a.y1 , so that w Ž a. w Ž b . s ¨ y1 f Ž ab. s w Ž ab.. Case 2. x 1 s a, x 2 s b, x 1 x 2 s w, a, b g M10 , w g Z1. By Definition 2.6Žiii., 2.3Ži., 2.6.2Žii., and 2.7, we have
w Ž ab . s w Ž w . s f Ž wby1 . f Ž by1 .
y1
s f Ž a . uy1 f Ž b .
s Ž ¨ y1 f Ž a . .Ž ¨ y1 f Ž b . . s w Ž a . w Ž b . . Case 3. x 1 s a, x 2 s w, a g M10 , w g Z1. As aw g M10 , w g Z1 , we have, by Definition 2.6 and 2.6.2,
w Ž aw . s ¨ y1 f Ž aw . s Ž ¨ y1 f Ž a . . Ž f Ž a .
y1
f Ž aw . . s w Ž a . w Ž w . .
PRODUCT-EQUALITY-PRESERVING MAPPINGS
663
Case 4. x 1 s w, x 2 s a, w g Z1 , a g M10 . As wa s aw, w Ž w . w Ž a. s w Ž a. w Ž w ., this case follows from Case 3. Case 5. x 1 s w 1 , x 2 s w 2 , w 1 , w 2 g Z1. Let a g M10 . As w 1 , w 2 , w 1w 2 y1 0 s w 2 w 1 g Z1 , we have wy1 2 g Z1 , w 1 a, w 2 a g M1 . Hence, by 2.6.2,
w Ž w1 . w Ž w 2 . s Ž f Ž w1 a. f Ž a.
y1
s f Ž w 1 a . f Ž wy1 2 a.
y1
y1 s f Ž w 1w 2 Ž wy1 2 a. . f Ž w2 a.
2.10.
y1
y1 . ž f Ž w2 Ž wy1 / 2 a. . f Ž w2 a.
y1
s w Ž w 1w 2 . .
w : G1 ª G 2 is an epimorphism mapping M1 onto ¨ y1 M2 .
Proof. By 2.8 and 2.9, it is left to show that w maps G1 onto G 2 . By Definition 2.6Žii. and 2.2Ži., M20 : w Ž M10 .. As Ž M20 . 2 s G 2 , w maps G1 onto G 2 by 2.9. The proof of Theorem 2.0}hence of Theorem 2}is complete.
ACKNOWLEDGMENT We are grateful to Ya. G. Berkovich for introducing us to the problem.
REFERENCES wBx wBFPx wBFx wF1x wF2x wF3x wFSx wJx
Ya. G. Berkovich, ‘‘Set Squaring in Groups,’’ to appear. Ya. G. Berkovich, G. A. Freiman, and C. Praeger, Small squaring and cubing properties for finite groups, Bull. Austral. Math. Soc. 44 Ž1991., 429]450. L. V. Brailovsky and G. a. Freiman, Groups with small cardinality of the cubes of their two-element subsets, Ann. New York Acad. Sci. 410 Ž1983., 75]82. G. A. Freiman, ‘‘Foundation of a Structural Theory of Set Addition,’’ Translations of Mathematical Monographs, Vol. 37, Am. Math. Soc., Providence, RI, 1973. G. a. Freiman, On two- and three-element subsets of groups, Aequationes Math. 22 Ž1981., 140]152. G. A. Freiman, Nonclosed semigroups with cancellation, Ann. New York Acad. Sci. 410 Ž1983., 91]98. G. A. Freiman and B. M. Schein, Interconnections between the structure theory of set addition and rewritability in groups, Proc. Am. Math. Soc. 113 Ž1991., 899]910. N. Jacobson, Basic Algebra, 2nd. ed., Freeman, New York, 1985.
182, 653]663 Ž1996.
0194
On Product-Equality-Preserving Mappings in Groups Gadi Moran Department of Mathematics and Computer Science, The Uni¨ ersity of Haifa, Haifa 31905, Israel Communicated by Leonard Lipshitz Received March 13, 1995
Let G, H be groups, M : G. A mapping f : M ª H is called a Product-equalitypreser¨ ing ŽPEP. mapping iff it satisfies ;a, b, c, d g M w ab s cd « f Ž a . f Ž b . s f Ž c . f Ž d . x .
Ž). M
THEOREM. Let G1 , G 2 be nonabelian groups with centers Z1 , Z2 and let M1 , M2 satisfy Gi _ Zi : Mi : Gi ,
i s 1, 2.
Let f : M1 ª M2 be a PEP mapping that maps M1 onto M2 . Then there is an epimorphism w : G1 ª G 2 and ¨ g Z2 such that for every x g M1 , f Ž x . s ¨w Ž x . . Q 1996 Academic Press, Inc.
0. INTRODUCTION 0.0. Parameters of the multiplication table of a subset M of a group G often make a useful research tool. Typically, the number < M 2 < of distinct entries of this table for finite M is such a parameter. On one hand, an extensive body of research deals with determination of the possible structure of M when < M 2 < is small in specific groups, e.g., the additive group of integers ŽwF1x, wF3x.. On the other hand, interesting classification theorems for groups G, where < M 2 < - < M < 2 for every M : G of a fixed finite cardinality m, are available. See wF2x for m s 2, wBFPx for m s 3, and wBx, wBFx, and wFSx for related topics and further references. In comparing multiplication tables of different group subsets M, M9, a bijection between M and M9 that, along with its inverse, preserves product equality Ži.e., a PEP bijection with PEP inverse. plays an important role. Such a mapping is called an isomorphism by Freiman in wF1x and a 653 0021-8693r96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
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GADI MORAN
2-isomorphism in wFSx. See wBx for further remarks and open questions concerning this notion. A convenient setting for a systematic treatment of this line of research is provided by a category that we shall denote PEP Žsee wJx, Vol. II, Chapter 1, for basics of category theory.. The objects of PEP are ordered pairs Ž M, G ., where M is a subset of the group G. A PEP morphism from Ž M, G . to Ž M9, G9. is any PEP mapping f : M ª M9. It will be convenient to use M alone to denote the object Ž M, G . whenever the context allows. Let us make some observations. We let G, H, K denote groups in the sequel, and use w , c to denote group homomorphisms. 0.1. If M1 : G, M2 : H, w : G ª H is a homomorphism, and w Ž M1 . : M2 , then f [ w < M1 , the restriction of w to M1 , is a PEP mapping from M1 to M2 . 0.2. For h g H, let h LŽ h R . denote left Žright. multiplication by h in H. That is, h L , h R : H ª H are defined by h LŽ y . s hy,
h R Ž y . s yh
Ž ygH..
Let h9, h0 g H. If M : H satisfies hXL < M s hYR < M,
Ž 0.1.
then f [ hXL < M s hYR < M is a PEP mapping. Indeed, if a, b, c, d g M and ab s cd, then f Ž a . f Ž b . s hXLŽ a . hYR Ž b . s h9abh0 s h9cdh0 s hXLŽ c . hYR Ž d . s f Ž c . f Ž d . . 0.3. Let M1 : G, M2 : H, h9, h0 g H, and let w : G ª H be a homomorphism. We call f : M1 ª M2 the standard Ž h9, w , h0 .-PEP mapping iff f s hXL ( w < M1 s hYR ( w < M1 .
Ž 0.2.
That is, f Ž x . s h9w Ž x . s w Ž x . h0 for all x g M1. By 0.1 and 0.2, f is indeed a PEP mapping whenever f is the standard Ž h9, w , h0 .-PEP mapping. We shall call f the standard Ž h, w .-PEP mapping if h s h9 s h0 and f is the Ž h9, w , h0 .-PEP mapping. Thus, f : M1 ª M2 is the standard Ž h, w .PEP mapping if f Ž x . s hw Ž x . s w Ž x . h
Ž x g M1 . .
Ž 0.3.
Notice that if f : M1 ª M2 is the standard Ž h9, w , h0 .-PEP mapping and for some x g M1 , h9w Ž x . s w Ž x . h9, then in fact h9 s h0 and f is actually the standard Ž h, w .-PEP mapping, where h s h9 s h0.
PRODUCT-EQUALITY-PRESERVING MAPPINGS
655
We say that f : M1 ª M2 is a standard PEP mapping if it is the standard Ž h9, w , h0 .-PEP mapping for some h9, w , h0. 0.4. The composition of standard PEP mappings is a standard-PEP mapping. Indeed, if f : M1 ª M2 is the standard Ž h9, w , h0 . mapping, M3 : K, and g: M2 ª M3 is the standard Ž k9, c , k0 . mapping, then gf : M1 ª M3 is the standard Ž k9c Ž h9., cw , c Ž h0 . k0 .-PEP mapping. Indeed, for x g M1 , gf Ž x . s g Ž h9w Ž x . . s k9c Ž h9w Ž x . . s Ž k9c Ž h9 . . cw Ž x . , gf Ž x . s g Ž w Ž x . h0 . s c Ž w Ž x . h0 . k0 s cw Ž x . Ž c Ž h0 . k0 . . Similarly, if f : M1 ª M2 is the standard Ž h, w . mapping, g: M2 ª M3 is the standard Ž k, c . mapping, and k c Ž h. s c Ž h. k, then gf is the standard Ž k c Ž h., cw .-PEP mapping. 0.5. If f : M1 ª M2 is the standard Ž h1 , w , h 2 .-PEP mapping, w : G ª H is a group isomorphism Žso that f is one to one., and f maps M1 onto M2 , then fy1 : M2 ª M1 is also a PEP mapping. In fact, it is the standard Ž wy1 Ž hy1 ., wy1 , wy1 Ž hy1 ..-PEP mapping, as one readily checks. 1 2 Similarly, if f : M1 ª M2 is the standard Ž h, w .-PEP mapping, w : G ª H is an isomorphism Žso that f is one to one., and f maps M1 onto M2 , then the bijection fy1 : M2 ª M1 is also a PEP mapping, and, in fact, it is the standard Ž wy1 Ž hy1 ., wy1 .-PEP mapping. 0.6. Let M : G satisfy ;a, b, c, d g M w ab s cd « a s c & b s d x .
Ž 0.4. M
Then any f : M ª H is a PEP mapping, as Ž).M holds trivially. It follows that the inverse of a PEP bijection is not, in general, a PEP mapping. Ž0.4.M holds, for instance, when M freely generates G or when M s Ž a, y .: y g Y 4 is a set of Ždistinct. transpositions, all moving the element a, and G is a group of permutations of a set containing a4 j Y, including M. In contrast with 0.6, we have the following two theorems. The first theorem was proved by Freiman wF3x. THEOREM 1. Let G, H be groups. Then for e¨ ery PEP mapping f on G into H there is a homomorphism w : G ª H and ¨ g H centralizing w Ž M . such that f Ž x . s ¨w Ž x .
Ž xgM..
656
GADI MORAN
Let us say that M : G is cocentral in G iff G _ ZŽ G . : M, where ZŽ G . [ ¨ g G: ; x g Gw ¨ x s x¨ x4 is the center of G. Our main result is: THEOREM 2. Let G1 , G 2 be nonabelian groups and let Mi : Gi be cocentral in Gi , i s 1, 2. Then for e¨ ery PEP mapping f of M1 onto M2 there is an epimorphism w : G1 ª G 2 and ¨ g ZŽ G 2 . such that f Ž x . s ¨w Ž x .
Ž x g M1 . .
Theorems 1 and 2 are proved in Sections 1 and 2, respectively. The argument for Theorem 1 in Section 1 is the one we used in June 1993, en route toward the proof of Theorem 2 and unaware of Freiman’s work wF3x. It is included here for the reader’s convenience, as we feel it is a useful introductory vehicle for the more delicate argument presented for Theorem 2 in Section 2. Corollaries 0.7. By Theorem 1, the group of PEP bijections of a group G is the subgroup of SG , the symmetric group on the set G, generated by the group AutŽ G . of automorphisms of the group G and the set ¨ L : ¨ g ZŽ G .4 of Žleft. translations by central elements of G. Thus it is a subgroup of the holomorph HolŽ G . generated in SG by AutŽ G . and the set of left translations g L : g g G4 Žsee wJx, Vol. I, 1.10.. In particular: 0.8. If G is abelian, then the group of PEP bijections of G is HolŽ G .. 0.9. If G is nonabelian with center Z, then for every cocentral M in G, the group of PEP bijections of M is isomorphic to the group of standard PEP bijections of G mapping M onto M. In fact f ¬ f < M is such an isomorphism, by Theorem 2.
2. PROOF OF THEOREM 1 We assume in this section that G, H are groups and f : G ª H. We call f a PEP mapping iff Ž). G is satisfied, i.e., ;a, b, c, d g G ab s cd « f Ž a. f Ž b . s f Ž c . f Ž d . . We let 1 denote the identity element of G.
Ž 1.0.
PRODUCT-EQUALITY-PRESERVING MAPPINGS
657
1.1. The following are equi¨ alent: Ži. Žii. Žiii.
f is a PEP mapping, i.e., Ž1.0. holds. f Ž abcy1 . s f Ž a. f Ž b . f Ž c .y1 for all a, b, c g G. f Ž ab. f Ž1. s f Ž a. f Ž b . for all a, b g G.
Proof. Ži. « Žii.: Assume Ž1.0. and let a, b, c g G. As Ž abcy1 . c s ab, we have f Ž abcy1 . f Ž c . s f Ž a. f Ž b . and so Žii. holds. Žii. « Žiii.: Assume Žii. and let a, b g G. By Žii., f Ž ab. s f Ž ab1y1 . s f Ž a. f Ž b . f Ž1.y1 and so Žiii. holds. Žiii. « Ži.: Assume Žiii. and let a, b, c, d g G satisfy ab s cd. Then by Žiii., f Ž a . f Ž b . s f Ž ab . f Ž 1 . s f Ž cd . f Ž 1 . s f Ž c . f Ž d . . Thus, Ži. holds. 1.2. i.e.,
Let f : G ª H be a PEP mapping. Then f Ž1. g H centralizes f Ž G ., f Ž g . f Ž 1. s f Ž 1. f Ž g .
for all g g G.
Ž 1.1.
Proof. Indeed, by 1.1Žiii., f Ž g . f Ž 1. s f Ž 1 ? g . f Ž 1. s f Ž 1. ? f Ž g . . 1.3.
Let f : G ª H be a PEP mapping and define w : G ª H by
w Ž g . [ f Ž 1.
y1
fŽ g..
Then w Ž xy . s w Ž x . w Ž y . for all x, y g G, i.e., w is a homomorphism. Proof. By 1.1 and 1.2, we have
w Ž xy . s f Ž 1 .
y1
f Ž xy . s f Ž 1 .
y2
s f Ž 1.
y2
f Ž x . f Ž y . s Ž f Ž 1.
f Ž 1 . f Ž xy . s f Ž 1 . y1
f Ž x . .Ž f Ž 1 .
y2
y1
f Ž xy . f Ž 1 .
f Ž y. . s w Ž x. w Ž y. .
Proof of Theorem 1. Let f : G ª H be a PEP mapping. Define w : G ª H, ¨ g H, by
w Ž x . s f Ž 1.
y1
f Ž x. ,
¨ s f Ž 1. .
Then f Ž x . s ¨ w Ž x . and by, 1.2 and 1.3, w is a homomorphism and ¨ centralizes w Ž G .. Theorem 1 is proved.
658
GADI MORAN
2. PROOF OF THEOREM 2 We assume in this section that G1 , G 2 are nonabelian groups with centers Z1 , Z2 , Mi is a cocentral subset of Gi , i.e., Gi _ Zi : Mi : Gi , i s 1, 2, and f : M1 ª M2 is a PEP mapping of M1 onto M2 , i.e., ;a, b, c, d g M1 ab s cd « f Ž a. f Ž b . s f Ž c . f Ž d . .
Ž 2.0.
Under these assumptions, we prove: THEOREM 2.0. There is an epimorphism w : G1 ª G 2 of G1 onto G 2 and an element ¨ g Z2 such that ; x g M1 f Ž x . s ¨ w Ž x . .
Ž 2.1.
Let us put Mi0 s Gi _ Zi , so that Mi0 : Mi , i s 1, 2. We proceed through a series of propositions. The first lists useful properties of the sets Mi0 . 2.1. Ži. Žii. Žiii. Živ.
Mi0 satisfies for i s 1, 2: a g Mi0 m ay1 g Mi0 . c g Mi0 « 'a, b g Mi0 w c s ab x. w g Zi , a g Gi « w a g Mi0 m aw g Mi0 x. Let a, b g Gi . Then ab g Zi « ab s ba.
Proof. Recall that a g Mi0 if ' x g Gw xa / ax x. Hence; Ži. Follows from xa / ax m ay1 x s ay1 Ž xa. ay1 / ay1 Ž ax . ay1 s y1 xa . Žii. Let b, c g Gi , cb / bc. Thus b, c g Mi0 and c s Ž cby1 . b. Let a s cby1 . Then c s ab and b g Mi0 . Also, a g Mi0 as by cb / bc we have ba s bcby1 / cbby1 s c s cby1 b s ab. Žiii. Indeed, Mi0 is a union of the cosets aZi in Gi other than Zi . Živ. Let a, b g Gi . As bŽ ab. by1 s ba, we see that ab s ba whenever ab commutes with b. Thus, if ab g Zi certainly ab s ba. 2.2. Ži. M20 : f Ž M10 .. Žii. There is a u g Z2 such that a, ay1 g M1 « f Ž a . f Ž ay1 . s f Ž ay1 . f Ž a . s u.
Ž 2.2.
Proof. Ži. Let c g M20 : M2 and let d g M2 satisfy cd / dc. As f maps M1 onto M2 , there are a, b g M1 such that f Ž a. s c and f Ž b . s d. By Ž2.0., ab / ba Želse cd s dc . so that a, b f Z1 , i.e., a, b g M10 . Hence c s f Ž a. g f Ž M10 . and we have M20 : f Ž M10 ..
PRODUCT-EQUALITY-PRESERVING MAPPINGS
Žii.
659
0 M10 / f , so let a0 g M10 . By 2.1Ži., ay1 0 g M1 as well. Put
u [ f Ž a0 . f Ž ay1 0 .. Let now a, ay1 g M1. As aay1 s ay1 a s a0 ay1 0 , we obtain, by Ž2.0., f Ž a . f Ž ay1 . s f Ž ay1 . f Ž a . s u and so Ž2.2. holds. We show that u g Z2 . Let a g M10 . By Ž2.2., we have f Ž ay1 . s uf Ž a.y1 s f Ž a.y1 u, so u commutes with f Ž a.y1 for all a g M10 . By 2.1Ži., u commutes with f Ž a. for every a g M10 , i.e., u centralizes f Ž M10 .. But M20 : f Ž M10 . by Ži., and so u centralizes M20 s G 2 _ Z2 , whence u g Z2 . 2.3. Ži. Žii. Žiii.
Let a, b, c g M1. We ha¨ e: ay1 g M1 « f Ž ay1 .y1 s uy1 f Ž a. s f Ž a. uy1 . abcy1 g M1 « f Ž abcy1 . s f Ž a. f Ž b . f Ž c .y1 . abc, c, cy1 g M1 « f Ž abc. s uy1 f Ž a. f Ž b . f Ž c ..
Proof. Ži. By 2.2Ži. f Ž a. f Ž ay1 . s f Ž a. f Ž ay1 . s u g Z2 , whence f Ž ay1 .
y1
s uy1 f Ž a . s f Ž a . uy1
Žii. By Ž2.0. and Ž abcy1 . c s ab, we have f Ž abcy1 . f Ž c . s f Ž a . f Ž b .
or
f Ž abcy1 . s f Ž a . f Ž b . f Ž c .
y1
.
Žiii. By Žii. and Ži., we have f Ž abc . s f ab Ž cy1 .
ž
y1
y1 y1
/ s f Ž a. f Ž b . f Ž c
.
s f Ž a . f Ž b . f Ž c . uy1 s uy1 f Ž a . f Ž b . f Ž c . . 2.4. There is a ¨ g Z2 such that a, b, ab, Ž ab .
y1
g M1 « ¨ s f Ž a . f Ž b . f Ž ab .
y1
s f Ž ab .
y1
f Ž a. f Ž b . .
Ž 2.3.
660
GADI MORAN
We split the proof of 2.4 into steps: 2.4.1.
Let a, b, ab, Ž ab.y1 , a, ˜ ˜b, ab, ˜˜ Ž ab ˜˜.y1 g M1. Then f Ž a. f Ž b . f Ž ab ˜˜. s f Ž ab . f Ž a˜. f Ž ˜b . .
Ž 2.4.
Proof. Indeed, let c s abab ˜˜ s abŽ ab ˜˜. s Ž ab. ab. ˜˜ By 2.3Žiii., f Ž c . s uy1 f Ž a . f Ž b . f Ž ab ˜˜. s uy1 f Ž ab . f Ž a˜. f Ž ˜b . . Ž2.4. follows. 2.4.2.
Let a, b, ab, Ž ab.y1 g M1. Then f Ž a . f Ž b . f Ž ab .
y1
s f Ž ab .
y1
f Ž a. f Ž b . .
Proof. Put a s a ˜ and b s ˜b in 2.4.1. 2.4.3.
Let a, b, ab, Ž ab.y1 , a, ˜ ˜b, ab, ˜˜ Ž ab ˜˜.y1 g M1. Then f Ž a . f Ž b . f Ž ab .
y1
s fŽa ˜. f Ž ˜b . f Ž ab ˜˜.
y1
.
Ž 2.5.
Proof. By 2.4.1, Ž2.4. holds. Hence f Ž ab .
y1
f Ž a. f Ž b . s f Ž a ˜. f Ž ˜b . f Ž ab ˜˜.
y1
.
By 2.4.2, Ž2.5. holds. 2.4.4.
Let a0 , b 0 , a0 b 0 g M10 and let ¨ [ f Ž a0 . f Ž b 0 . f Ž a0 b 0 .
y1
.
Then ¨ g Z2 . Proof. Let d g M20 . By 2.2Ži., there is a c g M10 with f Ž c . s d and, by 2.1Žii., there are a, b g M10 with c s ab. By 2.1Ži., cy1 s Ž ab.y1 g M10 and also Ž a0 b 0 .y1 g M10 . Thus, by 2.4.3 and 2.4.2 ¨ s f Ž a0 . f Ž b 0 . f Ž a0 b 0 .
y1
s f Ž a . f Ž b . f Ž ab .
y1
s f Ž ab .
y1
f Ž a. f Ž b . ,
i.e., ¨ s f Ž a. f Ž b . dy1 s dy1 f Ž a. f Ž b ., whence f Ž a. f Ž b . s d¨ s ¨ d. Thus, ¨ centralizes M20 , and so ¨ g Z2 . Proof of 2.4. As M10 / f , 2.2Ži. and Žii. guarantee the existence of a0 , b 0 g M10 such that also a0 b 0 g M10 , hence also Ž a0 b 0 .y1 g M10 . Use
PRODUCT-EQUALITY-PRESERVING MAPPINGS
661
a0 , b 0 to define ¨ g G 2 as in 2.4.4. By 2.4.1]2.4.4, ¨ g Z2 , and whenever a, b, ab, Ž ab.y1 g M2 , we have ¨ s f Ž a . f Ž b . f Ž ab .
2.5. Ži. Žii. Žiii.
y1
s f Ž ab .
y1
f Ž a. f Ž b . .
Let w g Z1 , aw, a, ay1 , bw, b, by1 g M1. Then: f Ž aw . f Ž a.y1 s f Ž bw . f Ž b .y1 . f Ž aw . f Ž a.y1 g Z2 . f Ž aw . f Ž a.y1 s f Ž a.y1 f Ž aw ..
Proof. Ži. As w s Ž aw . ay1 s Ž bw . by1 , Ž2.0. implies f Ž aw . f Ž ay1 . s Ž f bw . f Ž by1 .. By Ž2.2., f Ž ay1 . s f Ž a.y1 u and f Ž by1 . s f Ž b .y1 u, so that f Ž aw . f Ž a.y1 s f Ž bw . f Ž b .y1 . Žii. Let d g M20 . By 2.2Ži., let c g M10 satisfy d s f Ž c .. As cw, c, cy1 g M10 : M1 , we have, by Ži., f Ž aw . f Ž a.y1 s f Ž cw . f Ž c .y 1 . Thus df Ž aw . f Ž a.y1 s df Ž cw . f Ž c .y1 s f Ž c . f Ž cw . f Ž c .y1 . Hence, by 2.3Žii. and by w g Z1 , df Ž aw . f Ž a .
y1
s f Ž c ? cw ? cy1 . s f Ž cw . s f Ž wc . s f Ž Ž aw . ay1 c . s uy1 f Ž aw . f Ž ay1 . f Ž c . s uy1 f Ž aw . f Ž a .
y1
ud s f Ž aw . f Ž a .
y1
d,
where the last three equalities use 2.3Žiii., and 2.2Ži.. Thus, f Ž aw . f Ž a.y1 centralizes M20 , hence f Ž aw . f Ž a.y1 g Z2 . Žiii. f Ž aw . f Ž a.y1 s f Ž a.y1 f Ž aw . follows from Žii. by 2.1Živ.. We are now ready to define the epimorphism w : G1 ª G 2 and the element ¨ g Z2 mentioned in Theorem 2.0. DEFINITION 2.6. Let a0 , b 0 , a0 b 0 g M10 . Define an element ¨ g G 2 and a mapping w : G1 ª G 2 as follows: Ži. Žii. Žiii.
¨ [ f Ž a0 . f Ž b 0 . f Ž a0 b 0 .y1 .
w Ž a. [ ¨ y1 f Ž a. for a g M10 . w Ž w . [ f Ž a0 w . f Ž a0 .y1 for w g Z1.
Two remarks are in order: 2.6.1. ¨ is well defined, ¨ g Z2 , and satisfies Ž2.3.. Proof. M10 / f , so let c g M10 . By 2.1Žii., there are a0 , b 0 g M10 with a0 b 0 s c, and by 2.1Ži., also Ž a0 b 0 .y1 s cy1 g M10 . M10 : M1 , so ¨ s f Ž a0 . f Ž b 0 . f Ž a0 b 0 .y1 is well defined. By 2.4, 2.6.1 follows.
662
GADI MORAN
2.6.2. Ži. a g M10 « w Ž a. s ¨ y1 f Ž a. s f Ž a. ¨ y1 . Žii. w g M1 _ M10 , a g M10 « w Ž w . s f Ž aw . f Ž a.y1 s f Ž wa. f Ž a.y1 s f Ž a.y1 f Ž aw . s f Ž a.y1 f Ž wa. g Z2 . Proof. By 2.6.1 and 2.5. ¨ of Definition 2.6 and u of 2.2 are related in the next proposition.
2.7.
u s ¨ 2.
Proof. Let a, b, ab g M10 . Then also ay1, by1 , Ž ab.y1 g M10 and, by 2.6.1, 2.2, and 2.3Ži.,
2.8.
¨ 2 s f Ž a . f Ž b . f Ž ab .
y1
s f Ž a . f Ž b . f Ž ab .
y1
f Ž Ž ab .
y1 y1
.
f Ž by1 . f Ž ay1 .
uy1 f Ž ab . uf Ž b .
y1
uf Ž a .
y1
s u.
f Ž x . s ¨ w Ž x . for x g M1.
Proof. If x s a g M10 , 2.8 holds by Definition 2.6Žii.. Let x s w g M1 _ M10 s M1 l Z1 and let a g M10 . Then w, a, wa, Ž wa.y1 g M1 , so by Ž2.3., ¨ s f Ž w . f Ž a. f Ž wa.y1 and by 2.6.2, w Ž w . s f Ž wa. f Ž a.y1 . Hence ¨ w Ž w . s f Ž w .. 2.9.
w : G1 ª G 2 is a homomorphism.
Proof. Let x 1 , x 2 g G1. We show w Ž x 1 x 2 . s w Ž x 1 . w Ž x 2 .. Distinguish the following possible cases: Case 1. we have
x 1 s a, x 2 s b, a, b, ab g M10 . By Definition 2.6Žii. and 2.6.1,
w Ž a . w Ž b . s ¨ y1 f Ž a. ¨ y1 f Ž b . s ¨ y1 Ž ¨ y1 f Ž a. f Ž b . . and ¨ y1 s f Ž ab. f Ž b .y1 f Ž a.y1 , so that w Ž a. w Ž b . s ¨ y1 f Ž ab. s w Ž ab.. Case 2. x 1 s a, x 2 s b, x 1 x 2 s w, a, b g M10 , w g Z1. By Definition 2.6Žiii., 2.3Ži., 2.6.2Žii., and 2.7, we have
w Ž ab . s w Ž w . s f Ž wby1 . f Ž by1 .
y1
s f Ž a . uy1 f Ž b .
s Ž ¨ y1 f Ž a . .Ž ¨ y1 f Ž b . . s w Ž a . w Ž b . . Case 3. x 1 s a, x 2 s w, a g M10 , w g Z1. As aw g M10 , w g Z1 , we have, by Definition 2.6 and 2.6.2,
w Ž aw . s ¨ y1 f Ž aw . s Ž ¨ y1 f Ž a . . Ž f Ž a .
y1
f Ž aw . . s w Ž a . w Ž w . .
PRODUCT-EQUALITY-PRESERVING MAPPINGS
663
Case 4. x 1 s w, x 2 s a, w g Z1 , a g M10 . As wa s aw, w Ž w . w Ž a. s w Ž a. w Ž w ., this case follows from Case 3. Case 5. x 1 s w 1 , x 2 s w 2 , w 1 , w 2 g Z1. Let a g M10 . As w 1 , w 2 , w 1w 2 y1 0 s w 2 w 1 g Z1 , we have wy1 2 g Z1 , w 1 a, w 2 a g M1 . Hence, by 2.6.2,
w Ž w1 . w Ž w 2 . s Ž f Ž w1 a. f Ž a.
y1
s f Ž w 1 a . f Ž wy1 2 a.
y1
y1 s f Ž w 1w 2 Ž wy1 2 a. . f Ž w2 a.
2.10.
y1
y1 . ž f Ž w2 Ž wy1 / 2 a. . f Ž w2 a.
y1
s w Ž w 1w 2 . .
w : G1 ª G 2 is an epimorphism mapping M1 onto ¨ y1 M2 .
Proof. By 2.8 and 2.9, it is left to show that w maps G1 onto G 2 . By Definition 2.6Žii. and 2.2Ži., M20 : w Ž M10 .. As Ž M20 . 2 s G 2 , w maps G1 onto G 2 by 2.9. The proof of Theorem 2.0}hence of Theorem 2}is complete.
ACKNOWLEDGMENT We are grateful to Ya. G. Berkovich for introducing us to the problem.
REFERENCES wBx wBFPx wBFx wF1x wF2x wF3x wFSx wJx
Ya. G. Berkovich, ‘‘Set Squaring in Groups,’’ to appear. Ya. G. Berkovich, G. A. Freiman, and C. Praeger, Small squaring and cubing properties for finite groups, Bull. Austral. Math. Soc. 44 Ž1991., 429]450. L. V. Brailovsky and G. a. Freiman, Groups with small cardinality of the cubes of their two-element subsets, Ann. New York Acad. Sci. 410 Ž1983., 75]82. G. A. Freiman, ‘‘Foundation of a Structural Theory of Set Addition,’’ Translations of Mathematical Monographs, Vol. 37, Am. Math. Soc., Providence, RI, 1973. G. a. Freiman, On two- and three-element subsets of groups, Aequationes Math. 22 Ž1981., 140]152. G. A. Freiman, Nonclosed semigroups with cancellation, Ann. New York Acad. Sci. 410 Ž1983., 91]98. G. A. Freiman and B. M. Schein, Interconnections between the structure theory of set addition and rewritability in groups, Proc. Am. Math. Soc. 113 Ž1991., 899]910. N. Jacobson, Basic Algebra, 2nd. ed., Freeman, New York, 1985.