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Reports of the Academy of Sciences of the USSR
1966. Vol. 171, No. 4
UDC 512.86
MATHEMATICS
V. P. PLATONOV
THE FRATTINI SUBGROUP OF LINEAR GROUPS AND FINITARY APPROXIMABILITY
(Presented by Academician A. I. Mal’tsev on 19 II 1966)
Let \(G\) be an arbitrary group, and let \(\mathfrak M=(M_\alpha)\) be the set of all proper maximal subgroups of \(G\). The subgroup
\[ \Phi(G)=G\cap_\alpha M_\alpha \]
is called the Frattini subgroup of the group \(G\) (if the set \(\mathfrak M\) is empty, then \(\Phi(G)=G\)). According to the classical Frattini theorem, for a finite group \(G\) the subgroup \(\Phi(G)\) is nilpotent.
The question naturally arose of the possibility of extending this theorem to a broader class of groups, at least to finitely generated groups. The most general known result belongs to P. Hall \((^1)\): the Frattini subgroup of a finite extension of a finitely generated metanilpotent group is nilpotent. At the same time, in \((^1)\) an example is constructed of a soluble finitely generated group whose Frattini subgroup is not nilpotent. Although for arbitrary finitely generated groups the Frattini subgroup need not be nilpotent, in the case of linear groups over a field the question remained open. It was formulated explicitly by M. I. Kargapolov \((^2)\), problem 32).
In the present note a positive solution of this question is given for a wider class of linear groups than finitely generated ones*. The proof uses the simplest number-theoretic considerations and is based on approximating the groups under consideration by finite linear groups; for this purpose Mal’tsev’s approximation theorem \((^3)\) is somewhat generalized and refined; in particular, the projective homomorphisms of the group are naturally induced by homomorphisms of the corresponding commutative rings.
Let \(G\) be a group; let \((G_\alpha)\) be the set of its images under homomorphisms \(\varphi_\alpha:G\to G_\alpha\); let \(e_\alpha\) be the identity of \(G_\alpha\). If, for every \(e\ne g\in G\), \(\varphi_\alpha(g)\ne e_\alpha\) except for a finite number of indices \(\alpha\), then \(G\) will be called a limit of the groups \(G_\alpha\): \(G=\lim_{\to}(G_\alpha,\varphi_\alpha)\). We shall use analogous terminology also in the case of rings.
By \(P\) we denote the ground field, assumed arbitrary; \(P_0\) is the prime subfield. For arbitrary \(a_1,a_2,\ldots,a_k\in P\), by \(\Omega[a_1,a_2,\ldots,a_k]\) we denote the subring of \(P\) generated by \(1,a_1,a_2,\ldots,a_k\).
We shall need some elementary facts on approximation of commutative rings of finite type by fields.
If \(Z\) is the ring of integers and \(F\) is a finite field of characteristic \(q\), then by \(Z[X]=Z[x_1,x_2,\ldots,x_m]\), \(F[X]=F[x_1,\ldots,x_m]\) we denote the rings of commutative polynomials over \(Z\) and \(F\), respectively. Then
\[ Z[X]=\lim_{\to}(\Sigma_i,\varphi_i),\quad i=1,2,\ldots, \]
* It is known that in an arbitrary linear group the Frattini subgroup may fail to be nilpotent.
where \(\Sigma_i\) is a finite field of characteristic \(q_i\), \(q_i \ne q_j\) for \(i \ne j\);
\[ F[X]=\lim_{\longrightarrow}(\Delta_i,f_i), \]
where \(\Delta_i\) are finite fields of characteristic \(q\). In the second case the fields \(\Delta_i\) arise as the result of the natural homomorphisms
\[ f_i:\quad F[X]\to \Omega[a_1^{(i)},a_2^{(i)},\ldots,a_m^{(i)}], \]
where \(a_1^{(i)}, a_2^{(i)}, \ldots, a_m^{(i)}\) are elements of some finite extension \(F_i\) of the field \(F\). In the case of the ring of integers one considers the composite homomorphism
\[ \varphi_i:\quad Z[X]\to Z_{q_i}[X]\to \Omega[a_1^{(i)},a_2^{(i)},\ldots,a_m^{(i)}], \]
where \(Z_{q_i}\) is the residue field of \(Z\) modulo the prime \(q_i\). It is easy to see that any subring \(\Omega[a_1,a_2,\ldots,a_k]\subset P\) possesses an analogous approximation, for it is an algebraic extension of finite type of some polynomial ring \(Q[a_1,a_2,\ldots,a_t]\) over \(Z\) or \(F\). By \(d\) we denote the dimension of \(\Omega[a_1,a_2,\ldots,a_k]\) over the quotient field of the ring \(Q[a_1,a_2,\ldots,a_t]\).
Lemma 1. a) If \(P_0\) is a finite field, \(P_0^d\) is its extension of degree \(d\), then for any infinite algebraic extension \(R \supset P_0^d\)
\[ \Omega[a_1,a_2,\ldots,a_k]=\lim_{\longrightarrow}(\Omega_i,f_i), \]
where the finite fields \(\Omega_i\subset R\). b) If \(P_0\) is infinite; \(P_0^d\) is an extension of degree \(d\); \(p_1,p_2,\ldots,p_r\) are primes, then
\[ \Omega[a_1,a_2,\ldots,a_k]=\lim_{\longrightarrow}(\Omega_i,\varphi_i), \]
where the order of \(\Omega_i\) is equal to \(q_i^{dm_i}\), where \((m_i,p_j)=1\); \(i=1,2,\ldots\).
The proof follows from the preceding remarks, the countability of \(\Omega[a_1,a_2,\ldots,a_k]\), and the fact that a nonzero polynomial over an infinite field does not vanish identically. In case b) one considers infinite fields which are the union of finite fields of the required type.
Let \(\mathfrak R_1,\mathfrak R_2\) be commutative rings, and let \(GL(n,\mathfrak R_i)\) be the full linear group of degree \(n\) over \(\mathfrak R_i\), \(i=1,2\). Then a homomorphism \(\varphi:\mathfrak R_1\to\mathfrak R_2\) induces the natural homomorphism
\[ \Psi_\varphi:\quad GL(n,\mathfrak R_1)\to GL(n,\mathfrak R_2). \]
Theorem 1. Let \(\Gamma\) be a linear group of degree \(n\) over \(\Omega[a_1,a_2,\ldots,a_k]\). Then
\[ \Gamma=\lim_{\longrightarrow}(\Gamma_i,\Psi_{\varphi_i}), \]
where \(\Gamma_i\) are linear groups of degree \(n\) over finite fields \(\Omega_i\), which can be chosen as in Lemma 1.
Proof. Let \(\varphi_i:\Omega[a_1,a_2,\ldots,a_k]\to\Omega_i\) be the natural homomorphisms, where \(\Omega_i\) satisfy the requirements of Lemma 1. Let \(\Psi_{\varphi_i}\) be the homomorphisms of \(\Gamma\) into \(GL(n,\Omega_i)\) induced by \(\varphi_i\). For each \(e\ne g\in\Gamma\) there exists \(\Psi_{\varphi_i}\) such that \(\Psi_{\varphi_i}(g)\ne e\). Since the group \(\Gamma\) is countable, one constructs inductively such a sequence \((\Psi_{\varphi_i})\) that \(\Psi_{\varphi_i}(\Gamma)=\Gamma_i\) satisfy the condition
\[ \Gamma=\lim_{\longrightarrow}(\Gamma_i,\Psi_{\varphi_i}). \]
Corollary (3). A finitely generated linear group
\[ T=\lim_{\longrightarrow}(T_i,\Psi_i), \]
where \(T_i\) are finite linear groups of degree \(n\).
Indeed, \(T\) is contained in a linear group of degree \(n\) over \(\Omega[a_1,a_2,\ldots,a_k]\), where \(\Omega[a_1,a_2,\ldots,a_k]\) is generated by the elements of the generators and their inverses of the group \(T\).
We shall need two simple number-theoretic lemmas.
Lemma 2. Let \(p_1, p_2, \ldots, p_r, q\) be prime numbers. Then for the set \(S\) of numbers of the form
\[ s_i=\prod_{j=1}^{r}(p_j-1)q^i,\qquad i=1,2,\ldots, \]
there exist positive integers \(\alpha_1,\alpha_2,\ldots,\alpha_r\) such that
\[ \delta_i=\prod_{t=1}^{n}\left(q^{d s_i t}-1\right) =p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_r^{\alpha_r}\omega_i, \]
where \((p_j,\omega_i)=1,\ i=1,2,\ldots;\ j=1,2,\ldots,r\).
Proof. Let
\[ c=\prod_{j=1}^{r}(p_j-1), \]
\[ q^{d s_i k}-1=(q^{dck}-1)\left(\sum_{t=1}^{q_i}q^{dck(q^i-t)}\right). \]
Since \(p_j/(q^{p_j-1}-1)\), it follows that \(p_j/(q^{ck}-1)\). At the same time
\[ p_j\nmid \sum_{t=1}^{q^i} q^{dck(q^i-t)} \]
in view of the fact that \(p_j\nmid q\). If
\[ q^{dck}-1=p_1^{\beta_{1k}}p_2^{\beta_{2k}}\cdots p_r^{\beta_{rk}}\nu_k,\qquad (\nu_k,p_j)=1, \]
then
\[ \alpha_j=\sum_{k=1}^{n}\beta_{jk},\qquad j=1,2,\ldots,r. \]
Lemma 3. For any prime numbers \(p_1,p_2,\ldots,p_r\) there exists an infinite set \(\Theta\) of prime numbers and positive integers \(\alpha_1,\alpha_2,\ldots,\alpha_r\) such that for every \(m\) satisfying the condition \((m,p_j)=1,\ j=1,2,\ldots,r\),
\[ \delta_\theta=\prod_{t=1}^{n}\left(\theta^{dmt}-1\right) =p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_r^{\alpha_r}\omega_\theta, \]
where \((\omega_\theta,p_j)=1\), for any \(\theta\in\Theta\).
Proof. It is easily checked that the assertion of the lemma is valid if, as \(\Theta\), one takes the set of prime numbers of the form
\[ 1+p_1p_2\cdots p_r+sp_1^2p_2^2\cdots p_r^2,\qquad s=1,2,\ldots, \]
which is infinite by Dirichlet’s theorem on primes in an arithmetic progression.
Let now \(p_1,p_2,\ldots,p_r\ne q\) be all distinct prime divisors of the number \(n!\); \(q\) the characteristic of the ground field; \(q_1,q_2,\ldots\) an infinite sequence of prime numbers distinct from \(p_j,\ j=1,2,\ldots,r\); \(d\) some positive integer; \(\Delta_i\) a finite field whose order, for \(q>0\), is equal to \(q^{d(p_1-1)\cdots(p_r-1)q^i}\), and for \(q=0\) is equal to \(q_i^{dm_i}\), \((m_i,p_j)=1,\ i=1,2,\ldots\).
Lemma 4. If a linear group of degree \(n\)
\[ G=\lim_{\longrightarrow}(G_i,f_i), \]
where \(G_i\) are nilpotent linear groups of degree \(n\) over the fields \(\Delta_i\), which satisfy the conditions indicated above, then \(G\) is nilpotent.
Proof. It suffices to show that the nilpotency classes of the groups \(G_i\) are bounded in the aggregate. The group \(G_i=R_i\times U_i\), where \(R_i\) is a completely reducible subgroup and \(U_i\) is unipotent. Since the nilpotency class of \(U_i\) does not exceed \(n-1\), the groups \(G_i\) may be assumed completely reducible. If \(Z_i\) is the center of \(G_i\), then the order of the group \(G_i/Z_i\) has the form
\[ p_1^{\beta_1^{(i)}}p_2^{\beta_2^{(i)}}\cdots p_r^{\beta_r^{(i)}} \]
(see (4), p. 69). On the other hand, for \(q>0\) the order
\(G_i/Z_i\) is a divisor of the number
\[ \delta_i=\prod_{t=1}^{n}\left(q^{d s_i t}-1\right), \]
where \(s_i=(p_1-1)\ldots (p_r-1)q^i\), and for \(q=0\) it is a divisor of the number
\[ \gamma_i=\prod_{t=1}^{n}\left(q_i^{d m_i t}-1\right). \]
It follows from Lemmas 2 and 3 that there then exist such \(\alpha_1,\alpha_2,\ldots,\alpha_r\) that \(\beta_k^{(i)}<\alpha_k\) for all \(i=1,2,\ldots;\ k=1,2,\ldots,r\). The latter means that the orders of the groups \(G_i/Z_i\) are bounded in the aggregate, which entails the boundedness of the nilpotency classes.
The main result of this note is the following.
Theorem 2. Let \(G\) be a linear group of degree \(n\) over \(\Omega[a_1,a_2,\ldots,a_k]\). Then the Frattini subgroup \(\Phi(G)\) is nilpotent.
Proof. By Theorem 1,
\[ G=\lim_{\longrightarrow}(G_i,\Psi_i), \]
where \(G_i\) are linear groups of degree \(n\) over the finite fields \(\Delta_i\) from Lemma 4. Such a choice of the fields \(\Delta_i\) is possible by Lemma 1. If \(M_i\) is a maximal subgroup of \(G_i\), then \(H^{(i)}=\Psi_i^{-1}(M_i)\) is maximal in \(G\). We denote by \(\widetilde{\Phi}(G)\) the intersection of all such maximal subgroups of the group \(G\). Clearly, \(\Phi(G)\supseteq \widetilde{\Phi}(G)\). By construction,
\[ \widetilde{\Phi}(G)=\lim_{\longrightarrow}(\Phi(G_i),\widetilde{\Psi}_i), \]
where \(\widetilde{\Psi}_i\) are the restrictions of \(\Psi_i\) to \(\Phi(G)\). Since \(\Phi(G_i)\) are nilpotent, \(\widetilde{\Phi}(G)\) is also nilpotent by Lemma 4. The theorem is proved.
Corollary. The Frattini subgroup of a finitely generated linear group is nilpotent.
Remark 1. As is clear from the proof of Theorem 2, what has actually been proved is the nilpotency of the subgroup \(\widetilde{\Phi}(G)\supseteq \Phi(G)\), which is the intersection not of all maximal subgroups, but only of the maximal subgroups of finite index. In general, \(\widetilde{\Phi}(G)\ne \Phi(G)\).
Remark 2. In fact, Theorem 2 is true for any finitely generated commutative rings \(\Omega[a_1,a_2,\ldots,a_k]\) without nontrivial nilpotent elements, since under this condition \(\Omega[a_1,a_2,\ldots,a_k]\) is approximated by fields.
Belorussian State University
named after V. I. Lenin
Received
4 II 1966
REFERENCES
- P. Hall, Proc. London Math. Soc., 11, 327 (1961).
- The Kourovka Notebook. Unsolved Problems in Group Theory, Novosibirsk, 1965.
- A. I. Mal’tsev, Mat. sborn., 8, 405 (1940).
- D. A. Suprunenko, Solvable and Nilpotent Linear Groups, Minsk, 1958.