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UDC 519.45+512.86
MATHEMATICS
Yu. E. VAPNE
A CRITERION FOR REPRESENTABILITY OF A DIRECT WREATH PRODUCT OF GROUPS BY MATRICES
(Presented by Academician P. S. Novikov on 6 IV 1970)
The purpose of the present note is to prove the following two propositions.
Theorem 1. Let \(A, B\) be nontrivial groups isomorphically representable by matrices over a field of characteristic \(0\). The direct wreath product \(W=A\wr B\) is isomorphically representable by matrices over a field of characteristic \(0\) if and only if one of the following conditions is satisfied:
1) \(B\) is a finite group,
2) \(B\) is a finite extension of a torsion-free abelian group, and \(A\) is a torsion-free abelian group.
Theorem 2. Let \(A, B\) be nontrivial groups isomorphically representable by matrices over a field of characteristic \(p>0\). The direct wreath product \(W=A\wr B\) is isomorphically representable by matrices over a field of characteristic \(p\) if and only if one of the following conditions is satisfied:
1) \(B\) is a finite group,
2) \(B\) is a finite extension of a torsion-free abelian group, and \(A\) is an abelian \(p\)-group of finite period.
These propositions give the criterion mentioned in the title and thereby solve, for wreath products, question 3.29 from \((^{1})\). In paper \((^{2})\), Theorems 1, 2 were proved by the author for the case when the active group \(B\) is almost solvable. In the present note the general case is reduced to this particular case. Further, from Chevalley’s results on semisimple algebraic groups of matrices \(((^{3}), pp. 23–02)\) it follows that for matrix groups over a field, almost solvability is equivalent to the satisfaction of a nontrivial identity \((^{4})\). Therefore Theorems 1, 2 follow from \((^{2})\) and Lemmas 3, 4 proved below.
Let \(R\) be a ring without zero divisors, \(R[G]\) the group ring of the group \(G\) over \(R\). If \(m\) is a natural number and \(\gamma\) is an element of \(R[G]\) equal to
\[ \sum_{i=1}^{n} r_i g_i, \]
where \(r_i\) are nonzero elements of \(R\), \(g_i\in G\), then let \(Q(m,\gamma)\) denote the set of all possible products of at most \(m\) elements of the set
\[ \{g_2g_1^{-1},\; g_3g_1^{-1},\ldots,\; g_ng_1^{-1}\}. \]
We also denote
\[ Q(\gamma)=\bigcup_{m=1}^{\infty} Q(m,\gamma). \]
The sets \(Q(m,\gamma)\), \(Q(\gamma)\) in fact depend not on the element \(\gamma\), but on its expression. From the context it will always be clear which expression is meant.
Lemma 1. Let \(\alpha,\beta\in R[G]\), \(\alpha\ne 0\), and \(\alpha\beta=0\), and let \(m\) be the number of summands in an irreducible expression for \(\alpha\). Then the set \(Q(m,\beta)\) contains the identity (independently of which expression for \(\beta\) is fixed).
Proof. Let
\[ \alpha=\sum_i k_i a_i,\qquad \beta=\sum_j l_j b_j, \]
where \(k_i,l_j\) are nonzero elements of \(R\), \(a_i,b_j\in G\), and the elements \(a_i\) are pairwise distinct. By hypothesis,
\[ \alpha\beta=\sum_{i,j} k_i l_j a_i b_j=0. \]
Therefore
\[ a_i b_1=a_{\varphi(i)} b_{\psi(i)} \]
for
all \(i\), where \(\varphi,\psi\) are functions, and \(\psi(i)\ne 1\) for every \(i\). Hence
\[ a_{\varphi(i)}^{-1}a_i=b_{\psi(i)}b_1^{-1}. \tag{1} \]
Since \(\varphi\) maps the set \(\{1,2,\ldots,m\}\) into itself, there exist elements \(s,t\) of this set such that \(\varphi^s(t)=t\). Multiplying relations (1) for \(i=\varphi(t),\varphi^2(t),\ldots,\varphi^s(t)\), we obtain the identity on the left-hand side, and on the right an element of \(Q(m,\beta)\).
The lemma is proved.
Let \(F\) be the free group with free generators \(x_1,x_2,\ldots,\ldots,x_r,y_1,y_2,\ldots\). Let \(R\) be a ring with identity. Consider in \(R[F]\) the element
\[ \xi=\sum_{\pi}\operatorname{sgn}\pi\cdot x_{\pi(1)}x_{\pi(2)}\cdots x_{\pi(r)}, \tag{2} \]
where the summation is over all permutations \(\pi\) on \(r\) symbols, and \(\operatorname{sgn}\pi\) is the sign of the permutation \(\pi\).
Lemma 2. The set \(Q(\xi)\) does not contain the identity. In particular, no simple commutator of elements of the set \(Q(\xi)\), successively conjugated by the elements \(y_1,y_2,\ldots\), is equal to the identity.
Proof. We shall prove only the first assertion; the second is its immediate consequence.
Suppose, on the contrary, that some element
\[ x_{\pi_1(1)}\cdots x_{\pi_1(r)}x_r^{-1}\cdots x_1^{-1}\cdots x_{\pi_n(1)}\cdots x_{\pi_n(r)}x_r^{-1}\cdots x_1^{-1} \]
from \(Q(\xi)\), where \(\pi_1,\ldots,\pi_n\) are nonidentity permutations, is equal to the identity. This means that there exists a sequence of cancellations which transforms the above word into the empty word. At some step of this sequence the first occurrence of the element \(x_{\pi_1(i)}\), \(1\le i\le r\), must cancel with some occurrence of the element \(x_{\pi_1(i)}^{-1}\); then the word \(w_i\) standing between these occurrences must have become the empty word at preceding steps. It is easy to compute that \(\log w_i=\pi_1(i)-i\), where \(\log w_i\) is the sum of the exponents of all letters occurring in \(w_i\). But, as we have noted, \(w_i=1\), hence \(\log w_i=0\). Therefore \(\pi_1(i)=i\), i.e. the permutation \(\pi_1\) is the identity. The contradiction obtained proves the lemma.
Lemma 3. Let \(W=(a)\wr G\), where \((a)\) is an infinite cyclic group and \(G\) is a group. If the wreath product \(W\) is isomorphically representable by matrices over a field of characteristic \(0\), then a nontrivial identity holds on \(G\).
Proof. Suppose the wreath product \(W\) is isomorphically represented by matrices over an algebraically closed field \(k\) of characteristic \(0\).
First note that the matrix \(a\) may be assumed unipotent. If the group \(G\) is periodic, then by Schur’s theorem an identity holds on it ((\(^{5}\), p. 533)). Let \(G\) be nonperiodic and let \(g\) be an element of infinite order. The subgroup \((a,g)\) of the group \(W\) is isomorphic to the wreath product of two infinite cyclic groups and hence is solvable. By Mal’cev’s theorem ((\(^{5}\), p. 535)) the group \((a,g)\) is almost triangularizable; we shall assume it almost triangular. For some \(n\) the matrices \(a^n\) and \(g^n\) are triangular; therefore their commutator \(c=[a^n,g^n]\) is unipotent. By Lemma 12 from (\(^{2}\)) the subgroup \((c,G)\) is isomorphic to \(W\).
Thus, we shall assume that the matrix \(a\) is unipotent. Then the entire base \(F\) of the wreath product \(W\) is unipotent. There exists a nonzero vector \(v\) of the underlying space fixed by \(F\). Let \(U\) be the subspace spanned by the \(G\)-orbit of the vector \(v\). Obviously, \(U\) is invariant under \(G\) and fixed by \(F\), since
\[ vgf=vgfg^{-1}g=vf^{g^{-1}}g=vg,\quad \text{where } f\in F,\ g\in G. \]
This means that, with a suitable choice of basis of the underlying space, the elements of the group \(W\) have the form
\[ w=\begin{pmatrix} \varphi_{11}(w) & \varphi_{12}(w)\\ 0 & \varphi_{22}(w) \end{pmatrix}, \]
where \(\varphi_{22}(F)=e\), with \(e\) the identity matrix. Let \(K\) be the kernel of the homomorphism \(\varphi_{11}\). If \(K\) is trivial, induction on the degree of the matrices completes the proof. Let \(K\) be nontrivial. Since a nontrivial normal subgroup of a wreath product has nontrivial intersection with the base (see, for example, Lemmas 8.1 and 8.2 of \((^6)\)), it follows that \(K\cap F\ne 1\). Fix some nonidentity element \(f\) of \(K\cap F\). Obviously,
\(\varphi_{11}(f)=e,\ \varphi_{22}(f)=e\).
Let \(V\) be the linear space of all matrices over the field \(k\) having the same dimensions as the matrix \(\varphi_{12}(f)\). It is easy to see that the formula
\[ u\sum n_i g_i=\sum n_i\varphi_{11}(g_i)^{-1}u\varphi_{22}(g_i), \qquad n_i\in Z,\quad g_i\in G,\quad u\in V, \]
defines an action of the ring \(Z[G]\) on the space \(V\). This action determines a ring homomorphism
\(Z[G]\to \operatorname{End} V\), where \(\operatorname{End} V\) is the algebra of endomorphisms of the space \(V\) over the field \(k\). Let \(I\) be the kernel of this homomorphism.
We shall naturally regard the base \(F\) of the wreath product \(W\) as an exact cyclic right \(Z[G]\)-module (see, for example, \((^2)\)). Let \(f=a^\alpha,\ \alpha\in Z[G]\). By assumption \(\alpha\ne 0\). If \(\beta\in I\), then
\[ a^{\alpha\beta}=f^\beta= \begin{pmatrix} e & \varphi_{12}(f)\beta\\ 0 & e \end{pmatrix}=1, \]
whence \(\alpha\beta=0\). Thus, all elements of \(I\) annihilate \(\alpha\).
The factor ring \(Z[G]/I\) is isomorphically embedded in the algebra \(\operatorname{End} V\). Let \((r-1)\) be the dimension of this algebra over \(k\). According to \((^7,\) p. 329), the standard identity \(\xi=0\) holds on it, where \(\xi\) is the element from (2). Then
\(\beta=\xi(g_1,\ldots,g_r)\in I\) for any \(g_1,\ldots,g_r\in G\). As we noted above, \(\alpha\beta=0\). Let \(m\) be the number of terms in the reduced expression of the element \(\alpha\). By Lemma 1 the set \(Q(m,\beta)\) contains the identity. Let
\(q_1,\ldots,q_s\) be all the distinct elements of \(Q(m,\beta)\). Obviously, \(s\) depends only on \(f\) and \(r\), and
\(q_i=w_i(g_1,\ldots,g_r)\), where \(w_i\in Q(m,\xi)\). If \(h_1,\ldots,h_s\) are arbitrary elements of \(G\), then the simple commutator
\[ [q_1^{h_1},q_2^{h_2},\ldots,q_s^{h_s}]=1. \]
We have obtained an identity on the group \(G\). By Lemma 2 it is nontrivial.
Lemma 4. Let \(W=(a)\wr G\), where \((a)\) is a cyclic group of prime order \(p\), and let \(G\) be a group. If the wreath product \(W\) is isomorphically representable by matrices over a field of characteristic \(p\), then a nontrivial identity holds on \(G\).
Proof differs hardly at all from the proof of Lemma 3. It is only necessary to note that now the matrix \(a\) is unipotent by assumption (its order is equal to \(p\)), and instead of \(Z[G]\) one must everywhere consider the group ring over the prime field of characteristic \(p\).
In view of Lemma 3, Theorem 1 is now a consequence of the analogous Theorem 1 and Lemma 14 from \((^2)\). In exactly the same way, in view of Lemma 4, Theorem 2 is a consequence of Theorem 2 and Lemma 15 from \((^2)\). Let us note in passing that, in the proof of Lemma 15, the contradiction with the fact that the element \(a\) has infinite order is obtained more briefly and neatly as follows. Since the matrices \(a^b,\ b\in B\), belong to the diagonal group \(F_0\) and have the same eigenvalues as the matrix \(a\), the number of these matrices is finite. On the other hand, they are pairwise distinct in the wreath product \(W_0\)—a contradiction.
In conclusion, the author expresses gratitude to Yu. I. Merzlyakov for his constant attention and useful discussions.
Novosibirsk State
University
Received
26 III 1970
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