UDC 519.45

MATHEMATICS

M. A. BRONSTEIN

ON VERBAL SUBGROUPS OF FREE GROUPS

(Presented by Academician P. S. Aleksandrov, 26 I 1967)

In the present paper the following question is answered affirmatively: does the inclusion \(V(R) \supseteq V(S)\) imply the inclusion \(R \supseteq S\), if \(\mathfrak{B}\) is some variety of groups distinct from \(\mathfrak{E}\), the variety of all groups, and \(R\) and \(S\) are normal divisors of an arbitrary non-abelian free group \(F\). This assertion is a generalization of the well-known theorem of Auslander and Lyndon \((^{1})\) for the particular case of the variety of all abelian groups. Other special cases were considered by various authors \((^{2-5})\). Related questions were considered by T. Taylor and M. Danbud \((^{6})\). The history and bibliography of the question are presented most fully in \((^{5})\). In its ideas and methods the present work is close to \((^{5})\).

Let \(\mathfrak{B}\) be some variety of groups; let a group \(G\) be \(\mathfrak{B}\)-free and let the elements \(y_1, y_2, \ldots, y_m\) form its \(\mathfrak{B}\)-basis. We shall say that the elements \(y_1, y_2, \ldots, y_m\) multiply \(\mathfrak{B}\)-freely and that the group \(G\) is \(\mathfrak{B}\)-freely generated by the elements \(y_1, y_2, \ldots, y_m\).

Theorem. Let \(F\) be a non-abelian free group; let \(R\) and \(S\) be its normal divisors such that \(S \supset R\), and let \(\mathfrak{B}\) be a variety of groups such that \(V(S) \subseteq V(R)\); then \(\mathfrak{B}\) coincides with the variety \(\mathfrak{E}\) of all groups.

Obviously, this theorem is equivalent to the result formulated above.

In \((^{5})\) P. M. Neumann showed that it is enough to prove the theorem in the following special case: \(F\) is the free group of rank 2 with free basis \(\langle a,b\rangle\); \(R=\{a^p,b\}^F\), where \(p\) is a prime number; the normal divisor \(S\) is such that \(F=SR\). Hence it follows that in \(R\) there is an element \(z\) such that \(S \ni c=az\). Clearly, \(c=a^{pn+1}b^m w\), where \(w\in F'\).

Next, following the ideas of P. M. Neumann \((^{5})\), we construct a variety \(\mathfrak{D}\) having the following properties: \(D(R)=D(S)\); \(\mathfrak{D}\subseteq\mathfrak{B}\); \(\mathfrak{D}\) is generated by finite \(p\)-groups; hence, in particular, it follows that all free groups of this variety are approximated by finite \(p\)-groups. Obviously, it suffices for us to prove \(\mathfrak{D}=\mathfrak{E}\).

Lemma 1. Let \(H=F_k(\mathfrak{D})\) be the free group of the variety \(\mathfrak{D}\) of finite rank \(k\), and let \(y_1,y_2,\ldots,y_m\) be elements of the group \(H\), independent modulo the commutant \(H'\) and such that the factor group \(H/H'\{y_1,\ldots,y_m\}\) is a finite group of order relatively prime to \(p\). Then the elements \(y_1,y_2,\ldots,y_m\) multiply \(\mathfrak{D}\)-freely.

For the proof we refer to Lemma 2.4 of \((^{5})\).

Put \(A=F/D(R)\). We shall identify the elements of the group \(F\), as well as the normal divisors \(R\) and \(S\), with their images in \(A\). We know the following: \(A\) is generated by the elements \(a\) and \(b\); the elements \(a^p, a^{-i}ba^i\), \(0\leq i\leq p-1\), \(\mathfrak{D}\)-freely generate \(R\); the normal divisor \(S\) lies in \(\mathfrak{D}\) and contains the element \(c=az=a^{pn+1}b^m w\), where \(z\in R\), \(w\in A'\).

Lemma 2. In the group \(A\), the elements \(c^p, c^{-i}bc^i\), \(0\leq i\leq p-1\), multiply \(\mathfrak{D}\)-freely.

This lemma is close to Lemma 5.9 of \((^{5})\). One can prove that the elements under consideration satisfy the conditions of Lemma 2 in the \(\mathfrak{D}\)-free group \(R\).

Corollary 1. In the group \(A\), the elements \(c^p, b, [c^i,b]\), \(1\le i\le p-1\), are \(\mathfrak D\)-freely multiplied.

It suffices to indicate that \([c^i,b]=(c^{-i}bc^i)^{-1}b\).

Lemma 3. Let the group \(H\) be \(\mathfrak D\)-freely generated by elements \(x\) and \(y\). Then the elements \(x\) and \([x,y]\) also are \(\mathfrak D\)-freely multiplied.

Consider the homomorphism \(\varphi:H\to S\), defined by \(x\varphi=[c,b]\), \(y\varphi=c\). As is not hard to see, it is enough for us to prove that the elements \([c,b]\) and
\[ [[c,b],c]=[c,b]^{-1}[c^2,b][c,b]^{-1} \]
are \(\mathfrak D\)-freely multiplied. This is equivalent to saying that the elements \([c,b]\) and \([c^2,b]\) are \(\mathfrak D\)-freely multiplied. For \(p>2\) this follows directly from Corollary 1.

For \(p=2\) one should consider homomorphisms of the subgroup \(U\subset A\), \(\mathfrak D\)-freely generated by the elements \(c^2,b,[c,b]\), into the group \(S\), given by the maps
\([c,b]\to c^2,\ c^2\to[c,b],\ b\to c;\ [c,b]\to[c,b]^{-1},\ c^2\to c^2,\ b\to 1\), and
\([c,b]\to c,\ c^2\to[c,b],\ b\to 1\). It is easy to see that under the successive application of these homomorphisms the images of the elements \([c,b]\) and \([c^2,b]\) lie in \(U\) all the time. After applying all three homomorphisms, we see that the elements \([c,b]\) and \([c^2,b]\) go to the elements \([c,b], c^2\), which, by Corollary 1, are \(\mathfrak D\)-freely multiplied. Hence the assertion of the lemma follows for \(p=2\).

Lemma 4. In the group \(A\), the elements \(c\) and \([c,b]\) are \(\mathfrak D\)-freely multiplied.

Proof. Consider the images \(a^*, b^*, c^*\) of the elements \(a,b,c\) under the natural epimorphism \(A\to A^*=A/D(A)\). It is easy to see that the elements \(a^*,b^*\) \(\mathfrak D\)-freely generate \(A^*\), while the elements \(c^*\) and \(b^*\) satisfy the conditions of Lemma 1; then, by Lemmas 1 and 3, the elements \(c^*\) and \([c^*,b^*]\), to which the elements \(c\) and \([c,b]\) are mapped, are \(\mathfrak D\)-freely multiplied. This proves Lemma 4.

Further, the proof of the theorem is carried out analogously to the proof of Theorem 4.1 of the paper \((^5)\).

Consider the group \(H\), \(\mathfrak D\)-freely generated by elements \(x\) and \(y\), and in it the normal divisor \(R_1=\{x^p,y\}^H\). It, as a subgroup, is generated by the elements \(x^p,q_i,\ 0\le i\le p-1\), where \(q_i=x^{-i}yx^i\).

Lemma 5. In the group \(H\) the following holds: \(\alpha)\) for each \(i\), \(0\le i\le p-1\), the two elements \(x^p,q_i\) are \(\mathfrak D\)-freely multiplied; \(\beta)\) for each \(i\), \(0\le i\le p-1\), there exists an endomorphism \(\varepsilon_{1,i}:R_1\to R_1\) such that
\[ x^p\varepsilon_{1,i}=x^p,\quad q_i\varepsilon_{1,i}=q_i,\quad q_j\varepsilon_{1,i}=1,\quad 0\le j\le p-1,\quad i\ne j; \]
\(\gamma)\) the elements \(q_0\) and \(q_{p-1}\) are \(\mathfrak D\)-freely multiplied.

Proof. In view of Lemma 5 it is enough to prove the corresponding assertions for the elements \(c^p,z_i,\ 0\le i\le p-1\), where \(z_i=c^{-i}[c,b]c^i\). It is easy to see that these elements lie in the subgroup \(U\), \(\mathfrak D\)-freely generated by the elements \(c^p,c^{-i}bc^i,\ 0\le i\le p-1\). For each \(i\) construct a homomorphism \(\chi_i\) of the group \(U\) into the group \(H\), \(\mathfrak D\)-freely generated by elements \(x\) and \(y\), as follows:
\[ c^p\to x,\quad c^{-k}bc^k\to y,\quad k=0,1,\ldots,i, \]
\[ c^{-m}bc^m\to x^{-1}yx,\quad m=i+1,\ldots,p-1. \]
It is easy to see that \(c^p\chi_i=x,\ z_i\chi_i=[x,y],\ z_j\chi_i=1,\ i\ne j\). By Lemma 4, \(x\) and \([x,y]\) are \(\mathfrak D\)-freely multiplied. This proves \(\alpha)\).

Let \(\varphi_i\) be the restriction of \(\chi_i\) to \(\{c^p,z_1,\ldots,z_{p-1}\}=U\), and \(\psi_i\) to \(\{c^p,z_i\}\). Then \(\psi_i\) is an isomorphism and \(\operatorname{Im}\varphi_i=\operatorname{Im}\psi_i\). As is not hard to see, one may put \(\varepsilon_{1,i}=\varphi_i\psi_i^{-1}\).

To prove \(\gamma)\), construct the following homomorphism \(U\to H\):
\[ c^p\to x,\quad b\to y,\quad c^{-i}bc^i\to 1,\quad 1\le i\le p-1. \]
Under this homomorphism the elements \(z_0\) and \(z_{p-1}\) go to the elements \(y\) and \(x^{-1}y^{-1}x=[x,y]y^{-1}\), which are \(\mathfrak D\)-freely multiplied. Lemma 6 is proved.

Consider now in the group \(H\) the normal divisors \(R_n=\{x^{p^n},y\}^H\). Clearly, \(R_n\supseteq R_{n-1}\); moreover, \(R_n\), as a subgroup, is generated by the elements \(x^{p^n}, q_i,\ 0\le i\le p^n-1\), where \(q_i=x^{-i}yx^i\).

Lemma 6. In the group \(H\) the following holds: \(\alpha^*)\) for each \(i\), \(0\le i\le p^n-1\), the two elements \(x^{p^n},q_i\) are \(\mathfrak D\)-freely multiplied; \(\beta^*)\) for each \(i\), \(0\le i\le p^n-1\), there exists an endomorphism \(\varepsilon_{n,i}:R_n\to R_n\) such that,

that \(x^{p^n}\varepsilon_{n,i}=x^{p^n}\), \(q_i\varepsilon_{n,i}=q_i\), \(q_j\varepsilon_{n,i}=1\), \(0\leq j\leq p^n-1\), \(i\ne j\); \(\gamma^*)\) for all \(j\), \(0\leq j\leq p^{n-1}-1\), the two elements \(q_j\) and \(q_{j+(p-1)p^{n-1}}\) multiply \(\mathfrak D\)-freely.

The proof is by induction on \(n\) and coincides word for word with the proof of Lemma 4.3 of paper \((^5)\).

Put
\[ D_{n,j}=\{q_j,\ q_{j+(p-1)p^{\,n-1}}\},\quad 0\leq j\leq p^{n-1}-1. \]
From Lemma 6 and the definition of the subgroup \(D_{n,j}\) we obtain three corollaries:

Corollary 2. \(D_{n,j}\) is isomorphic to \(H\) for all \(n,j\), \(0\leq j\leq p^{n-1}-1\).

Corollary 3. \(D_{n,j}=x^{-j}D_{n,0}x^j\).

Corollary 4. For each \(n\), the subgroups \(D_{n,j}\) multiply properly (see (7)) over all \(j\), \(0\leq j\leq p^{n-1}-1\).

Corollaries 2 and 3 are trivial. To prove Corollary 4, we observe that \(\varepsilon_{n-1,j}\) acts identically on \(D_{n,j}\) and maps \(D_{n,k}\) to \(E\) when \(0\leq k\leq p^{n-1}-1\), \(j\ne k\).

Lemma 7. Suppose a group \(Q\) generates the variety \(\mathfrak B\). Suppose further that in some group \(K\), for each \(n\), there is a subgroup \(B_n\), an element \(d_n\), and natural numbers \(s_n\) such that: \(Q\approx B_n\); all the subgroups \(B_n, d_n^{-1}B_nd_n,\ldots,d_n^{-s_n}B_nd_n^{s_n}\) are distinct and multiply properly, and the numbers \(s_n\) are unbounded in the aggregate. Then \(\operatorname{var}(K)\supseteq \mathfrak B\circ\mathfrak A\), where \(\mathfrak A\) is the variety of all abelian groups.

This lemma generalizes Lemma 2.2 of paper \((^5)\) and can be proved in exactly the same way.

Apply Lemma 7 to the group \(H\), \(\mathfrak D\)-freely generated by the elements \(x\) and \(y\). For each \(n\) put \(B_n=D_{n,0}\), \(d_n=x\), \(s_n=p^{n-1}-1\). Corollaries 2, 3, 4 show that we are in the conditions of the lemma. Then we obtain \(\operatorname{var}(H)\supseteq \operatorname{var}(H)\circ\mathfrak A\), whence \(\operatorname{var}(H)=\mathfrak C\), the variety of all groups. But the group \(H\), by construction, lies in \(\mathfrak D\), therefore \(\mathfrak D=\mathfrak C\). The theorem is proved.

I express my gratitude to P. M. Neumann for posing the problem.

Moscow State University
named after M. V. Lomonosov

Received
20 I 1967

REFERENCES

\(^1\) M. Auslander, R. C. Lynodon, Am. J. Math., 77, 929 (1955).
\(^2\) H. Neumann, J. reine u. angew. Math., 212, 109 (1963).
\(^3\) H. Neumann, Arch. Math., 13, 1 (1962).
\(^4\) B. H. Neumann, Arch. Math., 13, 4 (1962).
\(^5\) P. M. Neumann, Arch. Math., 16, 6 (1965).
\(^6\) M. J. Dunwoody, Arch. Math., 16, 153 (1965).
\(^7\) O. N. Golovin, Matem. sborn., 27, (69), 427 (1960).