UDC 519.45

MATHEMATICS

E. M. LEVICH

ON A PROBLEM OF F. HALL

(Presented by Academician A. N. Tikhonov on 4 VII 1969)

One of the interesting and important problems in the theory of representations of groups is the problem of studying the properties of groups as a function of the properties of their representations. In particular, the following question arises: what properties must a group possess in order that all irreducible representations over an arbitrary field be finite-dimensional? It is known \((^{1-3})\) that, in order that all irreducible representations of a group \(\Gamma\) be finite-dimensional, it is sufficient that \(\Gamma\) have an abelian normal divisor of finite index and a finite number of generators. F. Hall \((^3)\) proved that every irreducible representation of a finitely generated nilpotent group over an absolutely algebraic field of prime characteristic \(p\) is finite-dimensional. On the other hand, in the same paper it is shown that if a polycyclic group \(\Gamma\) does not have an abelian normal divisor of finite index and a finite number of generators, and the field \(P\) is distinct from an absolutely algebraic field of prime characteristic \(p\), then \(\Gamma\) has an infinite-dimensional representation over the field \(P\).

The present note is devoted to the proof of a theorem asserting that every irreducible representation of a polycyclic group over an absolutely algebraic field of prime characteristic \(p\) is finite-dimensional. This theorem is an answer to the question posed by F. Hall in \((^3)\).

We shall use the following notation: \(\Gamma\) is a certain group; \(\gamma, \sigma, \varphi, \ldots\) are elements of the group \(\Gamma\); \(P\) is an absolutely algebraic field of prime characteristic \(p\); \(G\) is a vector space over the field \(P\); \(x, y, z, \ldots\) are elements of \(G\); \([\Gamma]\) is the group algebra of the group \(\Gamma\) over the field \(P\); \(R(\Gamma)\) is the locally nilpotent radical of the group \(\Gamma\); \(Z(R)\) is the center of the group \(R(\Gamma)\); \((G,\Gamma)\) is a representation of the group \(\Gamma\) by automorphisms of the vector space \(G\) over the field \(P\).

Let \(\Gamma\) be a polycyclic group, i.e. a group possessing a finite normal series with cyclic factors:

\[ \{e\}=\Gamma_0 \subset \Gamma_1 \subset \ldots \subset \Gamma_n=\Gamma . \tag{1} \]

We note that the group \(\Gamma\) is polycyclic if and only if \(\Gamma\) is a solvable group satisfying the maximal condition for subgroups \((^4)\).

Let \(I\) be a commutative domain of principal ideals which is not a field, and let \(\omega=\{p_\alpha\}\) \((\alpha\in\Lambda)\) be the set of all prime elements in \(I\). We shall say that an \(I\)-module \(V\) belongs to the class of \(I\)-modules \(\mathfrak{A}(I,\omega',k)\) \((\omega'\subset \omega,\ k\) a natural number) if and only if in \(V\) there is a free \(I\)-submodule \(V_0\) such that \(V/V_0\) is an \(\omega'\)-periodic \(I\)-module and the generator \(g\) of the order ideal \(\mathfrak{D}(v+V_0)\) \((v\in V,\ \mathfrak{D}(v+V_0)=\{h\in I:\ vh\in V_0\}=gI)\) of an arbitrary element \(v+V_0\in V/V_0\) can be represented in the form

\[ g=\prod_{\alpha\in\Lambda} p_\alpha^{\,n_\alpha}\quad(\alpha\in\Lambda), \]

where only a finite number of the \(n_\alpha\) are distinct from zero and all \(n_\alpha\leq k\).

An $I$-module $V$ is called $\omega'$-divisible ($\omega'\subset \omega$) if for any $v_0\in V$, $\lambda\in I$ the equation $v\lambda=v_0$ has a solution in $V$. If $V_0$ is a submodule of the $I$-module $V$, then we shall call it servile if from the fact that the equation $v\lambda=v_0$ ($v_0\in V$, $\lambda\in I$) has a solution in $V$ it follows that it has a solution in $V_0$.

Let $\sigma\in Z(R)$, and let $(G,\Gamma)$ be some representation of the group $\Gamma$, $I=P[x]$. Then it is clear that $G$ can be regarded as an $I$-module if the representation of $I$ relative to $G$ is given by the mapping $x\to\sigma$. For convenience, in this case we shall say that $G$ is an $I_\sigma$-module.

Lemma 1. Let $(G,\Gamma)$ be a cyclic representation of the polycyclic group $\Gamma$. Then every minimal servile $I$-submodule containing an element $x\in G$ belongs to the class $\mathfrak A(I_\sigma,\omega',k)$, where $\omega'$ is a subset of $\omega$.

Let $(G,\Gamma)$ be some representation of the polycyclic group $\Gamma$. Denote by
\[ G(\sigma_0,\lambda_0)=\{x\in G:\ x=y(\sigma_0-\lambda_0e),\ y\in G\} \]
($\sigma_0\in Z(R)$, $\lambda_0\in P$), and by $G(\sigma_1,\sigma_2,\ldots,\sigma_k,\lambda_1,\lambda_2,\ldots,\lambda_k)$ the linear span of the subspaces
\[ G(\sigma_1,\lambda_1),\ G(\sigma_2,\lambda_2),\ldots,\ G(\sigma_k,\lambda_k) \quad (\sigma_i\in Z(R),\ \lambda_i\in P). \]

It is clear that each $G(\sigma_i,\lambda_i)$ is a $[Z(R)]$-submodule in $G$.

Lemma 2. Let $(G,\Gamma)$ be a cyclic representation of the polycyclic group $\Gamma$, $\sigma_1,\sigma_2,\ldots,\sigma_m\in Z(R)$, $\lambda_1,\lambda_2,\ldots,\lambda_m\in P$. Then every minimal servile $[\sigma]$-submodule of the $[Z(R)]$-module
\[ G/G(\sigma_1,\sigma_2,\ldots,\sigma_m,\lambda_1,\lambda_2,\ldots,\lambda_m), \]
containing the element $x_0$, belongs to the class $\mathfrak A(I_\sigma,\omega',k)$, where $\omega'\subset \omega$ is a subset of primes.

From the preceding two lemmas the following proposition follows.

Main Lemma. Let $(G,\Gamma)$ be an irreducible representation of a polycyclic group over an absolutely algebraic field of prime characteristic $p$. Then in $G$ there exists a maximal proper subspace invariant with respect to $Z(R)$.

Theorem. Every irreducible representation of a polycyclic group $\Gamma$ by automorphisms of a vector space $G$ over an absolutely algebraic field of prime characteristic $p$ is finite-dimensional.

Proof.* Let $(G,\Gamma)$ be an exact irreducible representation of the group $\Gamma$. Then for any $x$ we have the equality $x[\Gamma]=G$. Let
\[ Q=\{g\in[\Gamma]:\ xg=0\}. \]
It is easy to see that $Q$ is a right ideal in $[\Gamma]$ and that the $[\Gamma]$-modules $G$ and $[\Gamma]/Q$ are isomorphic. According to the main lemma, in $G$ there is a maximal $Z(R)$-invariant subspace $H$. Since $Z(R)$ is an abelian group, the quotient space $G/H$ is one-dimensional (5), and this means that for each element $\sigma\in Z(R)$ there is such a $\lambda\in P$ that in the quotient $G/H$ the automorphism $\sigma$ induces the automorphism $\lambda e$ ($e$ is the identity automorphism). From the fact that $P$ is an absolutely algebraic field of prime characteristic $p$, it follows that for any $\sigma$ there is such a natural number $n$ that $\sigma^n$ induces the identity automorphism in the vector space $G/H$. Since $Z(R)$ is an abelian group with a finite number of generators, there is such a $k$ that $\sigma^k$ lies in the kernel of the representation $(G/H,Z(R))$ for any $\sigma\in Z(R)$. Denote by $Z^k(R)$ the subgroup in $Z(R)$ generated by the elements of the form $\sigma^k$, where $\sigma\in Z(R)$. It is easy to see that $Z^k(R)$ is a characteristic subgroup in $Z(R)$ and, consequently, a normal divisor of the group $\Gamma$. Taking into account that $(G,\Gamma)$ is an irreducible representation of the group $\Gamma$, we obtain that, by Remak’s theorem, $G$ is the direct sum of the spaces $H\gamma$ ($\gamma$ ranges over a set of representatives of all adjacent classes of the group $\Gamma$ with respect to the subgroup $Z(R)$). Since the group $Z^k(R)$ lies in the kernel of the representation $(G/H\gamma,Z(R))$ for any representative $\gamma$, it lies in the kernel of the repre—

* It is enough to consider the case of an algebraically closed field $P$.

representation \((G,\Gamma)\). Therefore \(Z(R)\) is a finite group, and then the group \(\Gamma\) is finite. Consequently, the vector space \(G\) is finite-dimensional. The theorem is proved.

Corollary. A polycyclic group \(\Gamma\) has a faithful irreducible representation over a field \(P\) if and only if it is finite.

Received
15 II 1968

REFERENCES

1. I. Kaplansky, Canad. J. Math., 1, 105 (1949).
2. S. Amitsur, Illinois J. Math., 5, No. 2, 198 (1961).
3. P. Hall, Proc. London Math. Soc. (3), 9, 595 (1959).
4. K. Hirsch, Proc. London Math. Soc., 49, 184 (1946).
5. B. I. Plotkin, Groups of Automorphisms of Algebraic Systems, Moscow, 1966.