MATHEMATICS

K. M. KUTYEV

PS-ISOMORPHISM OF AN ORDERED GROUP

(Presented by Academician A. I. Mal’tsev, 7 VII 1960)

In this paper it is proved that an ordered group is determined by the structure of its subsemigroups.

Let a one-to-one correspondence \(\varphi\) be established between the elements of two groups \(G\) and \(G^\varphi\). To an element \(g \in G\) there will correspond an element \(\varphi(g) \in G^\varphi\). Elements \(a, b\) of the group \(G\) will be called directly (respectively, inversely) \(\varphi\)-parallel if
\[ \varphi(ab)=\varphi(a)\varphi(b) \]
(respectively, if
\[ \varphi(ab)=\varphi(b)\varphi(a)). \]
If \(a, b\) are either directly or inversely \(\varphi\)-parallel, then they are called \(\varphi\)-parallel. Finally, if the elements \(a, b\) are (directly, inversely) \(\varphi\)-parallel and the elements \(b, a\) are (directly, inversely) \(\varphi\)-parallel, then they are called mutually (directly, inversely) \(\varphi\)-parallel.

Elements \(a\) and \(b\) of the group \(G\) will be called separable by a subsemigroup \(H\) if the element \(b\) belongs to \(H\), the element \(a\) belongs to the normalizer \(N(H)\) of the subsemigroup \(H\), and the intersection of the subsemigroup \(H\) with the cyclic subgroup \(\{a\}\) is equal to the identity.

Elements \(a\) and \(b\) of the group \(G\) will be called separable if there exists a subsemigroup \(H\) such that \(a\) and \(b\) are separable by the subsemigroup \(H\).

The product \(ab\) of elements \(a\) and \(b\) of the group \(G\) without torsion, having no other expression as a product of a finite number of positive powers of the elements \(a\) and \(b\), will be called unambiguous. In the contrary case the product \(ab\) is called ambiguous.

Let \(\varphi\) be an isomorphism of the structures of subsemigroups (a PS-isomorphism \((^1)\)) of the groups \(G\) and \(G^\varphi\). If \(A\) is a subsemigroup in \(G\), then \(A^\varphi\) is its image in \(G^\varphi\) under the PS-isomorphism \(\varphi\). \(\{A,B\}\) denotes the subsemigroup generated by the subsemigroups \(A\) and \(B\). In what follows we consider a group \(G\) without torsion. It is known that in this case the group \(G^\varphi\) is also without torsion and that between the elements of the groups \(G\) and \(G^\varphi\) one can establish a one-to-one correspondence \(\varphi\). To an element \(g \in G\) there will correspond an element \(\varphi(g) \in G^\varphi\), and
\[ \{g\}^{\varphi}=\{\varphi(g)\}. \]
For permutable elements \(a\) and \(b\) from \(G\) it is known:
\[ \varphi(ab)=\varphi(ba)=\varphi(a)\varphi(b)=\varphi(b)\varphi(a), \]
i.e. they are mutually \(\varphi\)-parallel \((^2)\).

A PS-isomorphism \(\varphi\) of a torsion-free group \(G\) onto a group \(G^\varphi\), under which any two separable elements \(a, b\) are mutually \(\varphi\)-parallel, will be called proper.

Lemma 1. Let the elements \(a\) and \(b\) of the torsion-free group \(G\) be nonpermutable and mutually \(\varphi\)-parallel. Then, if \(a, b\) are directly \(\varphi\)-parallel, then \(b, a\) are also directly \(\varphi\)-parallel.

Lemma 2. Let \(G\) be a torsion-free group. Let the elements \(a, b\) of the group \(G\) be separable by a commutative subsemigroup \(H\). Then the elements \(\varphi(a), \varphi(b)\) of the group \(G^\varphi\) are separable by the subsemigroup \(H^\varphi\), and the correspondence \(\varphi\) between the elements of the groups \(G\) and \(G^\varphi\) is a group isomorphism or anti-isomorphism of the subgroups \(H\) and \(H^\varphi\).

Lemma 3. Let \(a\) and \(b\) be noncommuting isolated elements of an \(R\)-group \(G\), and let at least one of the products \(ab\) or \(ba\) be unambiguous. Then \(a,b\) are isolated by a locally cyclic subgroup \(H\).

On the basis of Lemmas 2 and 3 the following theorem is proved.

Theorem 1. Let \(G\) be an \(R\)-group and let \(H\) be an isolated normal divisor in \(G\). Then \(H^\varphi\) is an isolated normal divisor in \(G^\varphi\).

Lemma 4. Let \(\varphi\) be a proper PS-isomorphism of the groups \(G\) and \(G^\varphi\); let \(H\) and \(F\) be isolated normal divisors in \(G\) and \(H\subset F\); let \(x\) and \(y\) be noncommuting elements, with \(x\in H,\ y\in F\setminus H\). Then \(\varphi(xy)=\varphi(x)\varphi(y)\) implies \(\varphi(xz)=\varphi(x)\varphi(z)\) for any \(z\in G\setminus F\).

Proof. It is enough to consider noncommuting elements \(x\) and \(z\). Let \(\varphi(xz)=\varphi(z)\varphi(x)\). 1) First we prove the lemma for any element \(z\in G\setminus F\) noncommuting with \(y\). Using the fact that the PS-isomorphism \(\varphi\) is proper, and Lemma 1, in exactly the same way as in \((^3)\), item (3), item (5) case 1 and item (6), we obtain a contradiction. 2) Now suppose \(zy=yz\). The equality \(\varphi(xy)=\varphi(x)\varphi(y)\), by item (3) of \((^3)\) and Lemma 1, implies
\[ \varphi(xyx)=\varphi(x)\varphi(y)\varphi(x)=\varphi(x)\varphi(yx). \]
The element \(yx\) does not commute with \(z\), since \(z\) does not commute with \(x\). The element \(yx\in F\setminus H\). Therefore the proof is the same as in item 1), where in place of the element \(y\) one must take the element \(yx\).

Lemma 5. Let \(\varphi\) be a proper PS-isomorphism of the groups \(G\) and \(G^\varphi\); let \(H\) and \(F\) be isolated normal divisors in \(G\) and \(H\subset F\); let \(f\) and \(g\) be noncommuting elements, with \(g\in F,\ f\in F\setminus H\). Then \(\varphi(fg)=\varphi(f)\varphi(g)\) implies \(\varphi(fh)=\varphi(f)\varphi(h)\) for any element \(h\in H\) noncommuting with \(g\).

Proof. It is enough to consider the case when \(f\) and \(h\) do not commute. The proof is literally the same as in item 1) of Lemma 4. In this proof one must put \(x=f,\ y=g,\ z=h\).

Theorem 2. Let \(\varphi\) be a proper PS-isomorphism of the groups \(G\) and \(G^\varphi\); let \(H\) and \(F\) be proper isolated normal divisors in \(G\) and \(H\subset F\). Then either any two elements \(h_1,h_2\in H\) are directly \(\varphi\)-parallel, or oppositely \(\varphi\)-parallel.

Proof. 1) Suppose that for any two elements \(x\) and \(y\) such that \(x\in H,\ y\in F\setminus H\), the equality \(\varphi(yx)=\varphi(x)\varphi(y)\) is valid; then we get:
\[ \varphi(h_1h_2)\varphi(y)=\varphi(yh_1h_2)=\varphi(h_2)\varphi(yh_1)=\varphi(h_2)\varphi(h_1)\varphi(y), \]
and the theorem is valid. 2) Suppose now that there is at least one pair of noncommuting elements \(x,y\) such that \(x\in H,\ y\in F\setminus H\), for which \(\varphi(yx)=\varphi(y)\varphi(x)\). Let \(z\) be any element such that \(z\in F\) and \(zy\ne yz\); then \(\varphi(zy)=\varphi(z)\varphi(y)\). Indeed, if \(\varphi(yzx)=\varphi(y)\varphi(zx)\), then, by Lemmas 1 and 4,
\[ \varphi(yz)\varphi(x)=\varphi(yzx)=\varphi(y)\varphi(zx)=\varphi(y)\varphi(z)\varphi(x), \]
and after cancellation, by Lemma 1, we obtain \(\varphi(zy)=\varphi(z)\varphi(y)\). If \(\varphi(yzx)=\varphi(zx)\varphi(y)\), then, by Lemmas 1 and 4,
\[ \varphi(z)\varphi(x)\varphi(y)=\varphi(zx)\varphi(y)=\varphi(yzx)\varphi(yz)\varphi(x)=\varphi(y)\varphi(z)\varphi(x). \]
(The last equality, in view of the fact that \(\varphi(yz)\varphi(x)=\varphi(z)\varphi(y)\varphi(x)\), would imply \(\varphi(z)\varphi(x)\varphi(y)=\varphi(z)\varphi(y)\varphi(x)\) and commutativity of \(x\) and \(y\), which is false.) Again after cancellation, by Lemma 1, we obtain \(\varphi(zy)=\varphi(z)\varphi(y)\). Put in Lemma 5 \(g=z,\ f=y\) and \(h=x\), if \(zy\ne yz\), and put \(g=z,\ f=yx\) and \(h=x\), if \(zy=yz\). We obtain, for any element \(x\) from \(H\) noncommuting with \(z\), \(\varphi(yx)=\varphi(y)\varphi(x)\) (or \(\varphi(yxx)=\varphi(yx)\varphi(x)\)). Now, by Lemmas 1 and 4, for any two noncommuting \(z\) and \(x\) such that \(x\in H,\ z\in F\), we obtain \(\varphi(xz)=\varphi(x)\varphi(z)\). Therefore for any elements \(x\) and \(z\), if \(x\in H,\ z\in F\), then \(\varphi(xz)=\varphi(x)\varphi(z)\). Further, in the same way as in item 1), we prove that for any \(h_1,h_2\) from \(H\) one has
\[ \varphi(h_1h_2)=\varphi(h_1)\varphi(h_2). \]

Let \(A\) be a semigroup from the group \(G\); let \(C\) be a subgroup of \(G\) containing \(A\), and let \(B\) be any semigroup from \(G\). Following B. I. Plotkin \((^4)\), we shall call a semigroup \(A\) semi-Dedekind if the equality
\[ \{A,B\}\cap C=\{A,B\cap C\} \]
holds.

Lemma 6. If \(I\) is an isolated subgroup in a torsion-free group \(G\), distinct from its normalizer \(N(I)\), it contains a semi-Dedekind sub-

semigroup \(A\) with identity, then the normalizer \(N(I)\) is contained in \(N(A)\), the normalizer of \(A\).

Proof. If \(x \in N(I)\setminus I\), then by Lemma 5.2 of \((^{4})\), \(x \in N(A)\). Now let \(x \in I,\ g \in N(I)\setminus I\); with the aid of Lemma 5.2 of \((^{4})\) we obtain
\[ A=x^{-1}g^{-1}Agx=x^{-1}Ax, \]
i.e. in this case also \(x \in N(A)\). Thus \(N(I)\subset N(A)\).

Lemma 7. Let, under a PS-isomorphism \(\varphi\), the image \(I^{\varphi}\) of an isolated normal divisor \(I\) of a group \(G\) be an invariant subgroup in \(G^{\varphi}\). Then the image \(A^{\varphi}\) of an invariant subsemigroup with identity \(A\), containing \(I\), is an invariant subsemigroup of \(G^{\varphi}\).

Proof. \(A\) is invariant in \(G\) and, by Lemma 5.1 of \((^{4})\), is a half-Dedekind subsemigroup in \(G\). But then \(A^{\varphi}\) is half-Dedekind in \(G^{\varphi}\) and, by Lemma 6, \(A^{\varphi}\) is invariant in \(G^{\varphi}\).

Theorem 3. Let \(G\) and \(G^{\varphi}\) be PS-isomorphic groups; let the group \(G\) be ordered, and let \(\Gamma\) be the subsemigroup of positive elements of the group \(G\). Then the group \(G^{\varphi}\) can be ordered, and as the subsemigroup of positive elements in \(G^{\varphi}\) one may take \(\Gamma^{\varphi}\).

Proof. First we prove the theorem in the case where \(G\) has a finite number of generators. From Lemma 1 of \((^{1})\) it is known that \(\Gamma^{\varphi}\) in \(G^{\varphi}\) is a linear subsemigroup with identity and without inverse elements. Thus it remains only to prove the invariance of \(\Gamma^{\varphi}\) in \(G^{\varphi}\). Let \(H\) be a maximal convex subgroup in \(G\); it is invariant and isolated in \(G\). The group \(G\) is ordered and therefore is an \(R\)-group. By Theorem 4, \(H^{\varphi}\) is an invariant subgroup in \(G^{\varphi}\). The PS-isomorphism of the groups \(G\) and \(G^{\varphi}\) induces a PS-isomorphism of the factor groups \(G/H\) and \(G^{\varphi}/H^{\varphi}\) \((^{5})\). \(G/H\) is abelian and, by \((^{2})\), \(G^{\varphi}/H^{\varphi}\) is also abelian. The subsemigroup \((\Gamma H)^{\varphi}=\Gamma^{\varphi}H^{\varphi}\) is linear in \(G\), and therefore
\[ \Gamma^{\varphi}H^{\varphi}=\Gamma^{\varphi}\cup H^{\varphi}. \]
It contains the commutant, since \(H^{\varphi}\subset \Gamma^{\varphi}H^{\varphi}\) and \(G^{\varphi}/H^{\varphi}\) is abelian. Therefore the subsemigroup \(\Gamma^{\varphi}\cup H^{\varphi}\) is invariant in \(G^{\varphi}\). By Lemma 9, the subsemigroup \((\Gamma\cap H)^{\varphi}=\Gamma^{\varphi}\cap H^{\varphi}\) is invariant in \(G^{\varphi}\). From the invariance of the subsemigroups \(\Gamma^{\varphi}\cup H^{\varphi}\), \(\Gamma^{\varphi}\cap H^{\varphi}\), and the invariance of the subgroup \(H^{\varphi}\) in \(G^{\varphi}\), it follows that \(\Gamma^{\varphi}\) is invariant in \(G^{\varphi}\). The theorem is extended in an obvious manner to any ordered group \(G\).

Lemma 8. Let \(\varphi\) be a proper PS-isomorphism. Let the series
\[ E=H_0\subset H_1\subset\cdots\subset H_{\alpha}\subset\cdots\subset G \]
and
\[ E^{\varphi}=H_0^{\varphi}\subset H_1^{\varphi}\subset\cdots\subset H_{\alpha}^{\varphi}\subset\cdots\subset G^{\varphi} \]
be ascending normal series in the \(R\)-groups \(G\) and \(G^{\varphi}\) such that the factors \(H_{i+1}/H_i,\ H_{i+1}^{\varphi}/H_i^{\varphi}\) \((i=1,2,\ldots,\alpha,\ldots)\) are locally nilpotent torsion-free groups. Let \(\varphi\) be an isomorphism (respectively an anti-isomorphism) of the nonabelian subgroups \(H_1\) and \(H_1^{\varphi}\). Then \(\varphi\) is an isomorphism (respectively an anti-isomorphism) of the groups \(G\) and \(G^{\varphi}\). If \(H_1\) is abelian, then \(\varphi\) is an isomorphism or an anti-isomorphism of the groups \(G\) and \(G^{\varphi}\).

Theorem 4. A PS-isomorphism of two ordered groups \(G\) and \(G^{\varphi}\) is always proper.

Theorem 5. Let the groups \(G\) and \(G^{\varphi}\) be PS-isomorphic and let the group \(G\) be ordered. Then the groups \(G\) and \(G^{\varphi}\) are isomorphic, and the PS-isomorphism \(\varphi\) is a consequence of their group isomorphism or anti-isomorphism.

Proof. It is evidently sufficient to prove the theorem for an ordered group \(G\) with a finite number of generators. By Theorem 3 we order the group \(G^{\varphi}\), taking as the subsemigroup of positive elements the subsemigroup \(\Gamma^{\varphi}\). By Theorem 4, the PS-isomorphism \(\varphi\) is proper. Since \(G\) has a finite number of generators, all convex subgroups in it are invariant. Let \(H\subset F\) be two adjacent convex subgroups in \(G\); then, by Theorem 2, \(\varphi\) is either an isomorphism or an anti-isomorphism

groups \(H\) and \(H^\varphi\). The set of all convex subgroups containing \(H\) is an ascending normal series in \(G\) with torsion-free abelian factors. Accordingly, the set of images of the terms of this series is an ascending normal series in \(G^\varphi\) (Theorem 1). Under a PS-isomorphism, \(\Gamma\)-convex subgroups are mapped onto \(\Gamma^\varphi\)-convex ones \((^1)\). If \(\Gamma\) is an invariant subsemigroup of \(G\) with identity and without inverse elements, then the definitions of a \(\Gamma\)-convex and of a convex subgroup in the group \(G\) coincide, if as the subsemigroup of positive elements of the group \(G\) one takes the subsemigroup \(\Gamma\) \((^6)\). Therefore, if \(A\) and \(B\) are adjacent convex subgroups in \(G\), then \(A^\varphi\) and \(B^\varphi\) are adjacent convex subgroups in \(G^\varphi\). Consequently, all factors of the series in the group \(G^\varphi\) are torsion-free abelian. Hence, by Lemma 8, it follows that \(\varphi\) is an isomorphism or an anti-isomorphism of the groups \(G\) and \(G^\varphi\).

Received
12 XII 1959

CITED LITERATURE

\(^1\) K. M. Kutylev, UMN, 7, issue 2, 193 (1956).
\(^2\) R. V. Petropavlovskaya, Matem. sborn., 29 (71), 1, 62 (1951).
\(^3\) W. R. Scott, Proc. Am. Math. Soc., 8, No. 6, 1141 (1957).
\(^4\) B. I. Plotkin, Tr. Mosk. matem. obshch., 6, 300 (1957).
\(^5\) A. S. Pekelis, Izvestiya Vyssh. uchebn. zaved., Matematika, No. 1, 189 (1957).
\(^6\) P. G. Kontorovich, Uch. zap. Sverdlovsk. gos. univ., issue 19, 3 (1956).