Doklady Akademii Nauk SSSR
1958, Volume 118, No. 6

MATHEMATICS

A. I. KOSTRIKIN

ON THE LOCAL NILPOTENCE OF LIE RINGS SATISFYING THE ENGEL CONDITION

(Presented by Academician I. M. Vinogradov on 16 IX 1957)

As is known, the question of the local nilpotence of Lie rings with the \(n\)-th Engel condition has so far been solved only for particular values \(n \leq 7\) \((^{1})\). The purpose of the present note is to give a brief exposition of the proof of the following theorem.

Theorem A. An arbitrary Lie ring \(L\) satisfying the \(n\)-th Engel condition
\[ [uv^n]=[\ldots[[uv]v]\ldots v]=0 \]
and having characteristic zero or finite characteristic \(p>n+[n/2]\), is locally nilpotent.

By the characteristic of a ring is meant here the common order of all nonzero elements in its additive group (the prime number \(p\)). The assertion of Theorem A is in fact true for a Lie ring \(L\) in which, from the equality \(mu=0,\ u\in L,\ m\leq n+[n/2]\), it follows that \(u=0\).

We note that the main idea of the proof of Theorem A is stated explicitly in § 5 of \((^{1})\). The proof is based on several auxiliary propositions, among which Lemma 1 occupies the most important place. The assertion contained in it may be formulated as follows: the local nilpotence of a Lie ring \(L\) with the \(n\)-th Engel condition and characteristic \(p>n\) will be proved if one establishes the existence in it of an element \(c\ne0\) for which the identity
\[ [cu^\alpha c]=0,\quad \alpha=1,2,3, \]
holds for every \(u\in L\). Such an element \(c\) can be found only when \(p>n+[n/2]\); however, it is evident that the difficulties remaining for arbitrary \(p>n\), even if they are not purely technical, are in any case such that attempts to overcome them are hopeless.

Keeping essentially the notation of \((^{1})\), let us introduce certain simplifications essential for the subsequent exposition. We shall everywhere regard as established the local nilpotence of Lie rings with the \((n-1)\)-st Engel condition and a given characteristic. Assuming a Lie ring \(L\), satisfying the \(n\)-th Engel condition, to be locally regular (and wishing thereby to arrive at a contradiction), we may, using the passage to the quotient ring modulo the radical \(N\), always suppose that \(N(L)=0\). In particular, \(L\) will be a ring without center, and therefore isomorphic to the ring \(D_L\) of inner derivations. We shall simply identify \(L\) and \(D_L\), agreeing to understand everywhere by a Lie ring a set of elements (denoted by small Latin letters \(a,b,\ldots\)) closed with respect to the operations of addition and “bracket” multiplication
\[ [uv]=uv-vu. \]
The elements of the enveloping ring \(\mathfrak{A}_L\) will be denoted by capital Latin letters \(A,B,\ldots\). If
\[ A=a_1a_2\ldots a_m=0, \]
then this means that
\[ [uA]=[\ldots[[ua_1]a_2]\ldots a_m]=[ua_1a_2\ldots a_m]=0 \]
for every \(u\in L\). Conversely, from the relation
\[ [ua_1a_2\ldots a_m]=0, \]
identically in \(u\), it follows that
\[ A=a_1a_2\ldots a_m=0. \]
The ring \(L\) satisfies the \(n\)-th Engel condition when \(u^n=0\) for all \(u\in L\). If the characteristic of the ring \(p>n\), then for

for an arbitrary pair of elements \(u, v \in L\) the identities

\[ \sum_{0\leq i<n-1} v^{i}uv^{\,n-1-i}=0, \tag{1} \]

\[ \sum_{0\leq i<n-1}(-1)^{i+1}\binom{n}{i+1}v^{i}uv^{\,n-1-i}=0. \tag{2} \]

Definition. Let us denote by \(c_{(m)}\) a nonzero element of a Lie ring satisfying the identities

\[ c_{(m)}u^{\alpha}c_{(m)}=0,\qquad \alpha=0,1,2,\ldots,2m-1\quad (m\geq 1). \]

It turns out that the following fact holds.

Lemma 1. From the existence of an element \(c_{(2)}\) in a Lie ring \(L\) with the \(n\)-th Engel condition and of characteristic \(p>n\) (or \(p=0\)), its local nilpotency follows.

The proof of this lemma is rather cumbersome, though not difficult. It is readily verified that the element

\[ c'_{(m)}=[c_{(m)}a^{2m+1}c_{(m)}],\qquad m\geq 1, \]

also has the properties of the element \(c_{(m)}\). But for \(m>1\) a much stronger assertion is valid, namely that

\[ c'_{(m)}u^{2m}c'_{(m)}v^{2m}c'_{(m)}=0, \]

whatever elements \(a,u,v\) of \(L\) we take. After this it is already comparatively easy to show that the right-normed product

\[ [c'_{(m)}b^{2m+1}c'_{(m)}],\quad b\in L, \]

may be taken as the element \(c_{(m+1)}\). It is assumed that \(2m+5\leq p\). Performing a chain of successive transitions from \(c_{(m)}\) to \(c_{(m+1)}\), we then consider in \(L\) the ideal \(\mathfrak N\) generated by the element \(c_{(m_0)}\) with index

\[ m_0=[(n-1)/2]. \]

The latter is possible, since \(n<p\), and hence \(2m_0+3\leq p\).

With the aid of identity (1), the elements \(c_{(m_0)}u^{n-2}c_{(m_0)}\) and \(c_{(m_0)}u^{n-1}c_{(m_0)}\) can be expressed in terms of products \(A\cdot B\cdot C\), \(B=c_{(m_0)}u^{\alpha}c_{(m_0)}\), \(\alpha<n-2\), which, by the definition of the element \(c_{(m_0)}\), are equal to zero. Thus \(\mathfrak N^2=0\), which contradicts the condition \(N(L)=0\).

Lemma 2. A Lie ring \(L\) of characteristic \(p\), generated by generators \(x_i\), \(x_i^2=0\) \((i=0,1,\ldots,d;\ d\ \text{arbitrary})\), and satisfying the \(n\)-th Engel condition, \(n<p\), is nilpotent.

First of all, let us note that one may restrict oneself to the particular case in which the additional condition \([x_i x_j]=0\), \(i,j=1,2,\ldots,d\), is imposed on the generators of the ring \(L\). This assertion follows from a more general theorem, whose proof (quite transparent) we omit.

According to Lemma 1, in \(L\) it suffices to find an element \(c_{(2)}\). This can be obtained as follows.

1) If \(c^2=0\), but \([ca^3c]\neq 0\) for some \(a\in L\), then there is an element

\[ e=e_1+e_2+e_3,\qquad e_i^2=0, \]

for which also \([ce^3c]\neq 0\).

Indeed, any element \(a\) of \(L\) can be written in the form \(a=\sum a_i\), \(a_i^2=0\), since from the conditions \(h_1^2=0\) and \(h_2^2=0\) it follows that \([h_1h_2]^2=0\).

2) By direct verification one establishes that if \(c^2=cu^2cv^2c=0\), then

\[ [ce^3c]u^2[ce^3c]=0. \]

Consequently, one may put \(c_{(2)}=[ce^3c]\).

3) Let \(c^2=0\) and \(cx_i x_j c=0\) for all \(i,j>0\). Then

\[ T=c_0u^2c_0v^2c_0=0, \]

where

\[ c_0=[cb^3],\qquad b=x_\alpha+x_\beta+x_\gamma,\qquad \alpha,\beta,\gamma>0. \]

Here it is first shown that the identities

\[ T_1(u)=c_0u^2cb^3=0 \]

and

\[ T_2(u)=c_0u^2bcb^2=0 \]

hold. From symmetry considerations one may likewise conclude that the expressions

\[ \overline{T_1(u)}=b^3cu^2c_0 \]

and

\[ \overline{T_2(u)}=b^2cbu^2c_0 \]

are equal to zero. After this the required relation is obtained:

\[ T=\{T_1(u)-3T_2(u)\}v^2c_0-c_0u^2\{\overline{T_1(v)}-3\overline{T_2(v)}\}=0. \]

4) Define recursively the sequence of elements:

\[ c_0=[x_0x_i],\quad i>0,\qquad c_1=[c_0a_1^2x_0],\ldots,\qquad c_m=[c_{m-1}a_m^2x_0], \]

\[ a_k=x_{i_k}+x_{j_k},\qquad i_k,j_k=1,2,\ldots,d. \]

By induction on \(m\) the identities are verified: \(c_m x_0=x_0 c_m=0\), \([c_m b^3 c_m]=0\), \(b=x_\alpha+x_\beta+x_\gamma,\ \alpha,\beta,\gamma>0\).

Suppose now that \(c_m\ne0\), but \(c_{m+1}=0\), whatever elements \(a_k\) we choose in constructing the sequence \(\{c_i\}\). Then the identity
\(c_m x_i x_j c_m=0,\ i,j=0,1,\ldots,d\), holds.

5) The finiteness of the number of nonzero terms in the sequence \(\{c_i\}\) is proved as follows.
\(c_{m+1}=[c_0[x_0a_1^2][x_0a_2^2]\ldots[x_0a_{m+1}^2]]=\)
\[ =-[x_i x_0 y_1y_2\ldots y_{m+1}]. \]
The number of distinct \(y_k=[x_0a_k^2]\) does not exceed \(\binom d2\), since \(a_k=x_{i_k}+x_{j_k}\). By hypothesis, the Lie ring with the \((n-1)\)-th Engel condition is locally nilpotent. Consequently, for sufficiently large \(m\)
\[ y_1y_2\ldots y_{m+1}=\sum u_i^{\,n-1}v_i^{\,r_i}, \]
where \(u_i\) and \(v_i\) are elements of the Lie ring with generators
\(y_1,\ldots,y_{\binom d2}\). Clearly, \([x_0y_k]=0\). Therefore
\[ nx_0u_i^{\,n-1}= \sum_{0\le s\le n-1}u_i^s x_0u_i^{\,n-1-s}=\sum z_j^n=0, \]
i.e. \(c_{m+1}=0\), which was to be established.

The combination of assertions 1)—5) leads us to the conclusion that there exists in \(L\) an element \(c_{(2)}\).

Lemma 3. For any pair of elements \(u,v\) of a Lie ring \(L\) with the \(n\)-th Engel condition and of characteristic \(p>n+[n/2]\) (or \(p=0\)), the identity
\[ [uv^{n-1}]^2=0 \]
holds.

Proof. Put
\[ \alpha_i=(-1)^i\binom ni, \]
\[ \Delta_{\nu,\mu}= \begin{vmatrix} \alpha_\nu\ldots \alpha_{\nu+\mu}\\ \ldots \ldots \ldots\\ \alpha_{\nu+\mu}\ldots \alpha_{\nu+2\mu} \end{vmatrix} =(-1)^{(\nu+\mu\mid 2)(\mu+1)} \frac{ \binom{n+\mu}{\mu+1}\binom{n+\mu-1}{\mu+1}\ldots\binom{n-\nu+1}{\mu+1} }{ \binom{\nu+2\mu}{\mu+1}\binom{\nu+2\mu-1}{\mu+1}\ldots\binom{\mu+1}{\mu+1} }, \]
\[ \nu\ge0,\qquad \mu\ge0,\qquad \nu+\mu\le n. \]

Obviously, \(\Delta_{\nu,\mu}\ne0\) if the characteristic \(p>n+\mu\).

The assertion of the lemma follows from the equalities
\[ A_{i,j}=v^j uv^{\,n-1-i}+uv^{\,n-1-i}=0,\qquad 0\le j\le i\le n-1, \]
whose verification we shall now undertake. Since \(v^{n-1}uv^{n-1}=0\), we have \(A_{0,j}=0\). Suppose that the equality \(A_{i,j}=0\) has been proved for \(i<k\) and all \(j\), and we shall prove it for \(i=k\). Further, we shall assume that \(A_{k,j}=0\) if \(j<l\), i.e. it remains to establish this identity for \(j=l,l+1,\ldots,k,\ l\ge0\). For \(l=0\) the condition \(A_{k,j}=0,\ j<0\), is trivially fulfilled, so the induction is justified. Multiplying identity (2) on the left by \(v^k uv^{\,n-1-k-s}\) and on the right by \(v^s,\ s=0,1,\ldots,n-1-k\), we obtain the system of identities
\[ \sum_{0\le i\le k-l}\alpha_{l+1+i+s}A_{k,l+i}=0. \]

Consider separately two cases.

1) \(n-k\ge k-l+1\). Since \(\Delta_{l+1,k-l}\ne0\), from the first \(k-l+1\) equations of the system we find what is needed:
\[ A_{k,l}=\ldots=A_{k,k}=0. \]

2) \(n-k<k-l+1\). Then express \(A_{k,l},\ldots,A_{k,l+n-1-k}\) in terms of \(A_{k,l+r}\), \(r\ge n-k\), using the inequality \(\Delta_{l+1,n-1-k}\ne0\). Multiplying identity (2) on the left by \(v^s\) and on the right by \(v^{l+r-s}uv^{n-1-l-r}\), \(s=n-1-k,\ldots,1,0\), we obtain
\[ \sum_{0\le i\le n-k-1}\alpha_{2k+l-n+i+s}A_{k+i,l+r}=0. \]

From this system of \(n-k\) equations with \(n-k\) unknowns and determinant \(\Delta_{2k+2-n,\,n-1-k}\ne 0\), we find \(A_{k,\,l+r}=0\) for any \(r>n-k\). Therefore also
\(A_{k,\,l}= \cdots = A_{k,\,l+n-1-k}=0\). All determinants of the form \(\Delta_{\nu,\mu}\) that occurred in the course of the proof were such that \(\mu\) did not exceed the number \([n/2]\). The lemma is proved.

Theorem A is now obtained in an obvious way (see, for example, the proof of Theorem 6 in \((^1)\)).

Let us note that the restriction on the characteristic arising in the proof of Lemma 3 can perhaps be reduced to \(p>n+1\) (for small \(n\) this has been checked), if one tries to use the identities (1) and (2) more fully. However, such an improvement would not be essential, since the most important case \(p=n+1\) (connected with the weakened Burnside hypothesis on periodic groups for a prime exponent \(p\)) would still remain outside consideration. A possible way of solving this important problem consists in changing the method of proof of the existence of the element \(c_{(2)}\).

In conclusion, the author sincerely thanks I. R. Shafarevich for his attention to this work.

Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR

Received
12 IX 1957

CITED LITERATURE

1. A. I. Kostrikin, Izv. AN SSSR, ser. matem., 21, no. 4 (1957).