UDC 512.86

MATHEMATICS

Academician of the Academy of Sciences of the BSSR D. A. SUPRUNENKO

ON THE THEORY OF SOLVABLE LINEAR GROUPS

§ 1. Let \(\Delta\) be an arbitrary field, and let \(G\) be a maximal irreducible primitive solvable subgroup of \(GL(n,\Delta)\). The structure of the group \(G\) is conveniently studied by means of the invariant series

\[ G \supseteq V \supseteq A \supseteq F \supset (e), \tag{1} \]

where \(F\) is a maximal Abelian normal divisor of the group \(G\); \(V\) is the centralizer of \(F\) in \(G\); \(A/F\) is a maximal subgroup among the Abelian invariant subgroups of \(G/F\) contained in \(V/F\).

Below we shall need the following known properties of the series (1).

1) \(G\) has a unique maximal Abelian normal divisor \(F\) \((^{4,5})\).

2) \(F\) is the multiplicative group \(\Sigma^*\) of some extension \(\Sigma\) of the field \(\Delta\), and the degree \(m=\Sigma:\Delta\) divides the number \(n\), while \(\operatorname{char}\Delta\) does not divide the number \(r=n/m\) \((^4)\).

3) The order of the group \(A/F\) is equal to \(r^2\); the Sylow subgroups of \(A/F\) are elementary Abelian groups; the linear \(\Delta\)-envelope of the group \(A\) coincides with the algebra \(\Sigma_r\) \((^{3,4})\).

4) If \(aF\) is an element of order \(\delta\) of the factor group \(A/F\), then the index of the centralizer of the element \(a\) in \(A\) is equal to \(\delta\) \((^4)\). If \(B\) is such a subgroup of \(A\) that \(B \supset F\), then the rank of the linear \(\Sigma\)-envelope of the group \(B\) over \(\Sigma\) coincides with the order of \(B/F\) \((^4)\).

5) Let \(p\) be a prime divisor of the number \(r\), and let \(B/F\) be a \(p\)-subgroup of \(A/F\), invariant in \(G/F\). Then

\[ B=(c_1)(d_1)\ldots(c_\nu)(d_\nu)F, \]

\[ (c_j,d_j)=c_jd_jc_j^{-1}d_j^{-1}=\eta;\quad j=1,\ldots,\nu; \]

\[ (c_i,c_j)=(d_i,d_j)=1;\quad i\ne j \Rightarrow (c_i,d_j)=1, \tag{2} \]

where \(\eta\) is an element of order \(p\) in \(F\). The order of \(B/F\) is equal to \(p^{2\nu}\) \((^4)\).

6) Let \(r=p_1^{l_1}\cdots p_k^{l_k}\) be the canonical decomposition of the number \(r\), and let \(P_i/F\) be the Sylow \(p_i\)-subgroup of \(A/F\). Then for \(i\ne j\) the mutual commutator \((P_i,P_j)\) is equal to 1. Obviously, the order of \(P_i/F\) is equal to \(p_i^{2l_i}\), and \(P_i\) is a normal divisor of \(G\). On solvable linear groups see also \((^{1,2,6})\). On the properties of \(A\) see \((^7)\).

In the present paper it is proved that, for an arbitrary field \(\Delta\), the group \(A\) of the series (1) is uniquely determined by the group \(G\). For an algebraically closed field \(\Omega\) it is shown that the description of maximal irreducible primitive solvable subgroups of \(GL(n,\Omega)\) reduces to the case when \(n\) is a power of a prime number. A one-to-one correspondence is established between the conjugacy classes of maximal irreducible primitive solvable subgroups of \(GL(p^l,\Omega)\) (\(p\) a prime number \(\ne \operatorname{char}\Omega\)) and the conjugacy classes of maximal \(s\)-irreducible (see the definition below) solvable subgroups of the symplectic group \(Sp(2l,p)\).

§ 2. Lemma 1. Let \(P/F\) be a \(p\)-subgroup of a Sylow group of \(A/F\), and let \(B/F\) be a subgroup of the group \(P/F\). Then the index of the centralizer of \(B\) in \(P\) is equal to the order of the group \(B/F\).

Lemma 2. Let \(B/F\) be a normal divisor of \(G/F\) contained in \(P/F\), where \(P/F\) is a Sylow \(p\)-subgroup of \(A/F\). Then the group \(P/F\) is representable

in the form of a direct product \(P/F = B/F \cdot C/F\), where \(C/F\) is a normal divisor of \(G/F\) such that \((B,C)=1\).

Proof. Let \(C\) be the centralizer of \(B\) in \(P\). Since \(B\) and \(P\) are normal divisors of \(G\), \(C\) is also a normal divisor of \(G\). The center of the group \(B\) coincides with \(F\); hence \(B\cap C=F\), i.e. \(B/F\cap C/F=F/F\). By Lemma 1, \(P:C=B:F\). Since \(P:F=(P:C)(C:F)\), we have \((B:F)(C:F)=P:F\). It follows from this and from the preceding that \(P/F\) is the direct product of its subgroups \(B/F\) and \(C/F\).

§ 3. Theorem 1. In the group \(G\) there is only one subgroup \(A\) such that: (I) \(A/F\) is an abelian normal divisor of \(G/F\); (II) \(A/F\subseteq V/F\); (III) \(A/F\) is maximal among the subgroups of \(G/F\) having properties (I) and (II).

Proof. Let \(A\) and \(B\) be such subgroups of \(G\) that \(A/F\) and \(B/F\) have properties (I), (II), (III). We shall show that \(A=B\). Let \(p\) be an arbitrary prime divisor of the number \(r\), let \(P/F\) be a Sylow \(p\)-subgroup of \(A/F\), and let \(Q/F\) be a Sylow \(p\)-subgroup of \(B/F\). Put \(D/F=P/F\cap Q/F\). Since \(D/F\) is a normal divisor of \(G/F\), by Lemma 2
\(P/F=D/F\cdot U/F,\ Q/F=D/F\cdot W/F\), where the direct factors \(U/F\) and \(W/F\) are normal divisors of \(G/F\), and \((D,U)=(D,W)=1\). Consequently, \(UW/F\) is an abelian normal divisor of \(G/F\). \(D/F\cdot UW/F=C/F\) is also an abelian normal divisor of \(G/F\), and \(C/F\subseteq V/F\). Obviously, \(C/F\supseteq P/F,\ C/F\supseteq Q/F\). In view of (III), \(C/F=P/F,\ C/F=Q/F\). Hence \(P=Q,\ A=B\). The theorem is proved.

Theorem 2. Let \(A\) be a group from the series (1); let \(\mathfrak N\) be the normalizer of \(A\) in \(GL(n,\Delta)\); and let \(G_1\) be a subgroup of \(GL(n,\Delta)\) such that \(A\triangleleft G_1\). If \(G_1\) is conjugate to \(G\) in \(GL(n,\Delta)\), then it is conjugate to \(G\) in \(\mathfrak N\).

Proof. Let \(G_1=dGd^{-1}\), \(d\in GL(n,\Delta)\). Since \(F\) is the center of \(A\) and \(A\triangleleft G_1\), \(F\) is an abelian normal divisor of \(G_1\), and \(d^{-1}Fd\) is an abelian normal divisor of \(G\). By property 1) of the series (1), \(d^{-1}Fd\subseteq F\). Since \(F=\Sigma^*\), where \(\Sigma\) is an extension of degree \(m\) of the field \(\Delta\), we have \(d^{-1}\Sigma d=\Sigma\), \(d^{-1}Fd=F\). Consequently, \(F\) is a maximal abelian normal divisor of \(G_1\). Since \(A:F=r^2\), \(A/F\) is maximal among the abelian invariant subgroups of \(G_1/F\) contained in \(V_1/F\), where \(V_1\) is the centralizer of \(F\) in \(G_1\). The group \(dAd^{-1}/F\) has the same properties. By Theorem 1, \(dAd^{-1}=A\), i.e. \(d\in\mathfrak N\). The theorem is proved.

§ 4. We now turn to the subgroups \(P_i\) of the group \(A\) (see 6)). The group \(A\) and its subgroups \(P_i\) will be regarded as subgroups of \(GL(r,\Sigma)\) (see property 3)).

Lemma 3. The irreducible components of the group \(P_j\) are pairwise equivalent and absolutely irreducible; their degrees are equal to the number \(p_j^{l_j}\) (see 6)).

Lemma 4. In a suitable basis of the space \(\Sigma^r\), the matrices \(a\) of the group \(A\) take the form
\[ a=u_1\times\cdots\times u_k, \tag{3} \]
\(GL(p_j^{l_j},\Sigma)\), isomorphic to the group \(P_j\), and \(\times\) denotes the Kronecker product of matrices. \(P_j\) consists of matrices of the form
\[ E_{p_1^{l_1}}\times\cdots\times u_j\times\cdots\times E_{p_k^{l_k}},\quad u_j\in P^j . \tag{4} \]

Lemma 5. Let \(H\) be the normalizer of \(A\) in \(GL(r,\Sigma)\), and \(H_j\) the normalizer of \(P_j\) in \(GL(p_j^{l_j},\Sigma)\). Then \(H\) consists of all matrices of the form
\[ h=h_1\times\cdots\times h_k, \tag{5} \]
where \(h_j\in H_j\).

Proof. Since \(P_j\) is a normal divisor of the group \(H\), for \(u_j\in P^j,\ h\in H\), by virtue of (4) we have
\[ h\left(E_{p_1^{l_1}}\times\cdots\times u_j\times\cdots\times E_{p_k^{l_k}}\right)h^{-1} = E_{p_1^{l_1}}\times\cdots\times u'_j\times\cdots\times E_{p_k^{l_k}}, \]

where \(u_j'\in P^j\). Obviously, the mapping

\[ u_j\to u_j' \tag{6} \]

is an automorphism of the group \(P^j\). If \(c,d\in P_j,\ \lambda\in\Sigma\), then
\(h(c+d)h^{-1}=hch^{-1}+hdh^{-1},\ h\lambda ch^{-1}=\lambda hch^{-1}\).
Consequently, the mapping (6) can be extended to an automorphism \(\varphi_j\) of the linear \(\Sigma\)-envelope \([P^j]\) of the group \(P^j\). By Lemma 3, \([P^j]\) coincides with the algebra \(\Sigma_{p^j}^{\,l_j}\). Hence
\[ \varphi_j(u_j)=h_j u_j h_j^{-1},\quad h_j\in GL(p^{l_j},\Sigma),\quad h_j\in H_j. \]
Thus, for any \(h\in H\) there are \(h_1,\ldots,h_k\), where \(h_j\in H_j\), such that
\[ hxh^{-1}=gxg^{-1},\quad g=h_1\times\cdots\times h_k \]
for every \(x\in A\). It follows that
\[ h^{-1}g=f\in F,\quad h=\sigma h_1\times\cdots\times h_k,\quad \sigma\in\Sigma^*. \]
Obviously, \(\sigma h_1\in H_1\).

Theorem 3. Let \(\Omega\) be an algebraically closed field, and let
\[ n=p_1^{l_1}\cdots p_k^{l_k} \]
be the canonical factorization of the number \(n\), with \(\operatorname{char}\Omega\) not dividing the number \(n\). Then every maximal irreducible primitive solvable subgroup \(G\) of the group \(GL(n,\Omega)\), in a suitable basis of \(\Omega^n\), is representable in the form
\[ G=G_1\times\cdots\times G_k, \]
where \(G_j\) is a maximal irreducible primitive solvable subgroup of
\[ GL(p_j^{l_j},\Omega),\quad j=1,\ldots,k, \]
and \(\times\) denotes the Kronecker-product sign.

§ 5. Let now \(\Delta=\Omega\), and let \(n=p^l\), where \(\Omega\) is an algebraically closed field and \(p\) is a prime number \((p\ne \operatorname{char}\Omega)\). Then, in a suitable basis of \(\Omega^n\), the group \(A\) of the series (1) has the form (4)

\[ A=(a_1)(b_1)\cdots(a_l)(b_l)F, \tag{7} \]

\[ a_j=E_{p^{j-1}}\times c\times E_{p^{l-j}},\quad b_j=E_{p^{j-1}}\times d\times E_{p^{l-j}},\quad j=1,\ldots,l, \]

where
\[ c=\operatorname{diag}[1,\eta,\ldots,\eta^{p-1}],\quad \eta\in\Omega,\quad \eta^p=1,\quad \eta\ne 1, \]

\[ d= \begin{bmatrix} 00\ldots 01\\ 10\ldots 00\\ 01\ldots 00\\ \cdots\\ 00\ldots 10 \end{bmatrix}. \]

Consider the normalizer \(\mathfrak N\) of the group \(A\) in \(GL(p^l,\Omega)\). For \(x\in\mathfrak N\),

\[ xa_jx^{-1}=a_j'=\lambda_j a_1^{\alpha_{1j}}\cdots a_l^{\alpha_{lj}}b_1^{\gamma_{1j}}\cdots b_l^{\gamma_{lj}}, \]

\[ xb_jx^{-1}=b_j'=\mu_j a_1^{\beta_{1j}}\cdots a_l^{\beta_{lj}}b_1^{\delta_{1j}}\cdots b_l^{\delta_{lj}}, \tag{8} \]

\[ \lambda_j,\mu_j\in\Omega^*;\quad \alpha_{ij},\beta_{ij},\gamma_{ij},\delta_{ij}\in GF(p), \]

and \(a_j\) and \(b_j\) satisfy the conditions

\[ (a_i,b_i)=\eta;\quad (a_i,a_\mu)=1;\quad (b_i,b_\mu)=1;\quad i\ne\mu\Rightarrow (a_i,b_\mu)=1. \tag{9} \]

By virtue of (8), \(a_j'\) and \(b_j'\) satisfy the same conditions:

\[ (a_i',b_i')=\eta;\quad (a_i',a_\mu')=1;\quad (b_i',b_\mu')=1;\quad i\ne\mu\Rightarrow (a_i',b_\mu')=1. \tag{10} \]

Put

\[ \psi(x)= \begin{bmatrix} \alpha_{11}\ldots \alpha_{1l}\beta_{11}\ldots \beta_{1l}\\ \cdots\\ \alpha_{l1}\ldots \alpha_{ll}\beta_{l1}\ldots \beta_{ll}\\ \gamma_{11}\ldots \gamma_{1l}\delta_{11}\ldots \delta_{1l}\\ \cdots\\ \gamma_{l1}\ldots \gamma_{ll}\delta_{l1}\ldots \delta_{ll} \end{bmatrix}, \quad x\in\mathfrak N. \tag{11} \]

Conditions (10) are equivalent to the following equalities:

\[ 1.\quad \sum_{\rho=1}^{l}(\alpha_{\rho i}\delta_{\rho i}-\beta_{\rho i}\gamma_{\rho i})=1. \]

\[ 2.\quad \sum_{\rho=1}^{l}(\alpha_{\rho i}\gamma_{\rho\mu}-\alpha_{\rho\mu}\gamma_{\rho i})=0. \tag{12} \]

1. \[ \sum_{\rho=1}^{l}(\beta_{\rho i}\delta_{\rho\mu}-\beta_{\rho\mu}\delta_{\rho i})=0. \tag{12} \]

2. \(\mu\ne i\Rightarrow\)
   \[ \sum_{\rho=1}^{l}(\alpha_{\rho i}\delta_{\rho\mu}-\beta_{\rho\mu}\gamma_{\rho i})=0. \]

We introduce the symplectic group \(\operatorname{Sp}(2l,p)\) over \(GF(p)\) in the following way. Let
\[ \Phi_l=\begin{bmatrix} 0&E_l\\ -E_l&0 \end{bmatrix}. \]
The symplectic group \(\operatorname{Sp}(2l,p)\) will be the group of all matrices \(h\) in \(GL(2l,p)\) satisfying the condition
\[ {}^{t}h\Phi_l h=\Phi_l, \tag{13} \]
where \({}^{t}h\) is the transpose of \(h\).

Lemma 6. The elements of the matrix
\[ h= \begin{bmatrix} \alpha_{11}\ldots\alpha_{1l}\beta_{11}\ldots\beta_{1l}\\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ \alpha_{l1}\ldots\alpha_{ll}\beta_{l1}\ldots\beta_{ll}\\ \gamma_{11}\ldots\gamma_{1l}\delta_{11}\ldots\delta_{1l}\\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\ \cdot\\ \gamma_{l1}\ldots\gamma_{ll}\delta_{l1}\ldots\delta_{ll} \end{bmatrix} \]
over \(GF(p)\) satisfy conditions (12) if and only if \(h\in\operatorname{Sp}(2l,p)\).

Lemma 7. \(\operatorname{Sp}(2l,p)\simeq \mathfrak{R}/A\), for the mapping
\[ \psi:\mathfrak{R}\to\operatorname{Sp}(2l,p), \]
defined by formula (11), is an epimorphism whose kernel is the group \(A\).

Lemma 8. Let \(G_1\) and \(G_2\) be such maximal irreducible primitive solvable subgroups of \(GL(p^l,\Omega)\) that \(A\triangleleft G_1\), \(A\triangleleft G_2\). Then \(G_1\) and \(G_2\) are conjugate in \(GL(p^l,\Omega)\) if and only if \(H_1=\psi(G_1)\) and \(H_2=\psi(G_2)\) are conjugate in \(\operatorname{Sp}(2l,p)\).

A subgroup \(H\) of the group \(\operatorname{Sp}(2l,p)\) will be called \(s\)-reducible if there exist an integer \(\nu\), \(1\le \nu\le l\), and a matrix \(t\in\operatorname{Sp}(2l,p)\) such that the matrices of the group \(t^{-1}Ht\) have the form
\[ \begin{bmatrix} \alpha_{11}\ldots\alpha_{1\nu} & *\\ \cdot\ \cdot\ \cdot\ \cdot & \\ \alpha_{\nu1}\ldots\alpha_{\nu\nu} & \\ 0\ldots 0 & \\ \cdot\ \cdot\ \cdot\ \cdot & *\\ 0\ldots 0 & \end{bmatrix}. \]
In the opposite case the subgroup \(H\) of the group \(\operatorname{Sp}(2l,p)\) is called \(s\)-irreducible. The following is true.

Lemma 9. Let \(H\) be a subgroup of \(\operatorname{Sp}(2l,p)\). The subgroup \(\psi^{-1}(H)\) of the group \(GL(p^l,\Omega)\) is primitive if and only if \(H\) is irreducible.

From the last two lemmas it follows:

Theorem 4. Let \(H_1,\ldots,H_\rho\) be maximal solvable \(s\)-irreducible subgroups of \(\operatorname{Sp}(2l,p)\), and let every maximal solvable \(s\)-irreducible subgroup of \(\operatorname{Sp}(2l,p)\) be conjugate in \(\operatorname{Sp}(2l,p)\) with one and only one \(H_j\). Then
\[ \psi^{-1}(H_1),\ldots,\psi^{-1}(H_\rho) \]
are maximal solvable irreducible primitive subgroups of \(GL(p^l,\Omega)\), and every maximal solvable irreducible primitive subgroup of \(GL(p^l,\Omega)\) is conjugate with one and only one \(\psi^{-1}(H_j)\).

Institute of Mathematics
Academy of Sciences of the BSSR

Received
8 V 1968

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