UDC 512.86

MATHEMATICS

R. T. VOL’VACHEV

PERIODIC NILPOTENT LINEAR GROUPS OVER THE FIELD OF RATIONAL NUMBERS

(Presented by Academician A. I. Mal’tsev on V 22, 1965)

The theory of linear nilpotent and locally nilpotent groups was developed by D. A. Suprunenko \((^{1-6})\). He also studied periodic linear locally nilpotent groups over an algebraically closed field of characteristic zero \((^7)\).

In the present note we study periodic nilpotent subgroups of the full linear group \(GL(n,R)\) over the field of rational numbers \(R^*\). It is proved that two maximal periodic nilpotent subgroups of \(GL(n,R)\) are conjugate in \(GL(n,R)\) if their centers are conjugate in \(GL(n,R)\). It is also shown that in \(GL(n,R)\) there exists, up to conjugacy, only a finite number of maximal nilpotent periodic subgroups. A complete description is given of the maximal periodic nilpotent subgroups of \(GL(n,R)\).

1. We first consider irreducible nilpotent periodic subgroups of \(GL(n,R)\). Their structure is closely connected with the structure of linear \(p\)-groups over the field of rational numbers (see Theorem 1). (For results on linear \(p\)-groups see \((^9)\), where Sylow \(p\)-subgroups of the full linear group over an arbitrary field are studied.)

The structure of irreducible nilpotent periodic subgroups is described by the following theorem.

Theorem 1. Let \(\Gamma\) be an irreducible periodic nilpotent subgroup of \(GL(n,R)\). Then \(\Gamma\) is conjugate in \(GL(n,R)\) to a certain subgroup of the group of all matrices \(g\) of the form

\[ g = a_1 \times a_2 \times \cdots \times a_\nu, \tag{1} \]

where \(\times\) is the sign of the Kronecker product, \(a_i\) ranges over an irreducible \(p_i\)-subgroup of a Sylow subgroup of the group \(GL(\varphi(p_i^{\alpha_i}),R)\) (\(\varphi\) is Euler’s function). Consequently,

\[ n=\varphi(k), \tag{2} \]

where \(k=p_1^{\alpha_1}p_2^{\alpha_2}\ldots p_\nu^{\alpha_\nu}\) is the canonical decomposition of the number \(k\) (see (1)), and the group of all matrices of the form (1) is determined uniquely, up to conjugacy in \(GL(n,R)\), by specifying the number \(k\) from (2).

Corollary. If \(n\ne \varphi(k)\), where \(k\) is an integer and \(\varphi\) is Euler’s function, then in \(GL(n,R)\) there are no irreducible nilpotent periodic subgroups.

Now the following question naturally arises: which groups of matrices of the form (1) are maximal irreducible nilpotent periodic subgroups of \(GL(n,R)\), or, in other words, to which numbers \(k\) satisfying the condition \(n=\varphi(k)\) does there correspond a maximal irreducible nilpotent periodic subgroup of \(GL(n,R)\).

* We note that every locally nilpotent subgroup of \(GL(n,R)\) is nilpotent \((^{6,8})\).

Let us turn to the solution of this question. It is easy to see that the group of matrices of the form (1) can be regarded as a subgroup of \(GL(r,\Sigma)\), where

\[ \Sigma = R(\varepsilon), \tag{3} \]

\(\varepsilon\) is a primitive root of unity of degree \(m\), and (cf. (2))

\[ m=p_1p_2\cdots p_\nu, \tag{4} \]

\[ r=n/\varphi(m). \tag{5} \]

We shall first show that the group of all matrices of the form (1) is almost always maximal among irreducible nilpotent periodic subgroups whose linear \(R\)-envelope of the center coincides with \(\Sigma\). More precisely, the following is true.

Lemma 1. Let \(\Gamma\) be the group of all matrices of the form (1), determined by such a number \(k\) that \(n=\varphi(k)\), and let \(k=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_\nu^{\alpha_\nu}\) be the canonical factorization of the number \(k\), and let \(\Sigma=R(\varepsilon)\), where \(\varepsilon\) is a primitive root of unity of degree \(m=p_1p_2\cdots p_\nu\). Then \(\Gamma\) is maximal among irreducible nilpotent periodic subgroups of the group \(GL(n,R)\) whose linear \(R\)-envelopes of the centers coincide with \(\Sigma\), except for the case when \(\Gamma\) is determined by a number \(k\) satisfying the relation \(n=\varphi(k)=\varphi(2k)\).

Now one can answer the question formulated earlier. The answer is given by the following

Theorem 2. For any \(k\) such that \(n=\varphi(k)\), the group of all matrices of the form (1), determined by this number \(k\), is a maximal irreducible nilpotent periodic subgroup of \(GL(n,R)\), except for the case when the group (1) is determined by such a number \(k\) for which the relation \(n=\varphi(k)=\varphi(2k)\) is possible.

From Theorems 1 and 2 there follows the following

Theorem 3. Two maximal irreducible nilpotent periodic subgroups of \(GL(n,R)\) are conjugate in \(GL(n,R)\) if their centers are conjugate in \(GL(n,R)\).

The theorems formulated above give a complete classification and construction of maximal irreducible nilpotent periodic subgroups of \(GL(n,R)\). For convenience, let us combine the results obtained as follows.

First main theorem. If \(n\ne\varphi(k)\), where \(\varphi\) is Euler’s function and \(k\) is an integer, then in \(GL(n,R)\) there are no irreducible nilpotent periodic subgroups. Let \(\mathfrak M\) be the set of all such numbers \(k\) that \(n=\varphi(k)\), except for that number \(k_1\) for which the relation \(n=\varphi(k_1)=\varphi(2k_1)\) is possible, and let

\[ k=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_\nu^{\alpha_\nu}, \tag{6} \]

be the canonical factorization of the number \(k\).

Then:

1. For any \(k\) from \(\mathfrak M\), in \(GL(n,R)\) there is a maximal irreducible nilpotent periodic subgroup whose matrices can be represented in the form

\[ g=a_1\times a_2\times\cdots\times a_\nu, \]

where \(a_i\) runs through a \(p_i\)-subgroup of the Sylow group \(GL(\varphi(p_i^{\alpha_i}),R)\) (see (6)), and \(\times\) denotes the Kronecker product.

1. Different \(k\) from \(\mathfrak M\) determine nonisomorphic groups.

2. Every maximal irreducible nilpotent periodic subgroup of \(GL(n,R)\) is conjugate in \(GL(n,R)\) to the group determined by some \(k\) from \(\mathfrak M\).

Corollary. In \(GL(n,R)\), up to conjugacy, there exists only a finite number of maximal irreducible nilpotent periodic subgroups.

II. Let now \(\Gamma\) be a maximal nilpotent periodic subgroup of \(GL(n,R)\). As is known, any periodic subgroup of \(GL(n,R)\) is completely reducible \((^{13},\,^{14})\). Consequently, \(\Gamma\) is a completely reducible group.

Let \(G\) be one of the irreducible components of the group \(\Gamma\). If \(m\) is the degree of \(G\), then \(m=\varphi(k)\), and \(G\) is a maximal irreducible nilpotent periodic subgroup of \(GL(m,R)\). Consequently, a maximal periodic nilpotent subgroup of the group \(GL(n,R)\) is uniquely determined, up to conjugacy in \(GL(n,R)\), by the degrees \(n_1,n_2,\ldots,n_s\) of its irreducible components and by the centers of these irreducible groups, or, in other words, by a set of numbers \(k_1,k_2,\ldots,k_s\) such that, for every \(k_i\),

\[ \varphi(k_i)\ne \varphi(2k_i), \tag{7} \]

where the numbers \(n_i\), where

\[ n_i=\varphi(k_i), \tag{8} \]

satisfy the relation

\[ n=n_1+n_2+\cdots+n_s,\qquad n_i>0. \tag{9} \]

Thus, it is necessary to determine to what set of numbers \(k_1,k_2,\ldots,k_s\), satisfying (7), (8), and (9), there corresponds a maximal nilpotent periodic subgroup of \(GL(n,R)\). We shall formulate this assertion more precisely.

Let \(n\) be represented in the form (9), with each \(k_i\) from (8) satisfying (7). Represent the space \(R^n\), in which \(GL(n,R)\) acts, in the form

\[ R^n=\Sigma_1+\cdots+\Sigma_s, \tag{10} \]

where \(\Sigma_i\) is a subspace of dimension \(n_i\). By \(\Gamma_{k_i}\) we shall denote the subgroup of \(GL(n,R)\) which induces on each \(\Sigma_j\) \((j\ne i)\) the identity group, and on \(\Sigma_i\) the maximal irreducible nilpotent periodic group of degree \(n_i\) determined by the number \(k_i\), with \(n_i=\varphi(k_i)\).

Construct the group

\[ \Gamma_{k_1,k_2,\ldots,k_s}=\Gamma_{k_1}\Gamma_{k_2}\cdots\Gamma_{k_s}. \tag{11} \]

It turns out that the group \(\Gamma_{k_1,k_2,\ldots,k_s}\) is not always a maximal nilpotent periodic subgroup of \(GL(n,R)\). Our task is to find conditions under which the sets of numbers \(k_1,k_2,\ldots,k_s\), satisfying (7), (8), and (9), determine a maximal nilpotent periodic subgroup \(\Gamma_{k_1,k_2,\ldots,k_s}\) of the group \(GL(n,R)\). The method used by us for this goes back to \((^{15})\).

Lemma 2. If the formula

\[ i\ne j\Rightarrow k_i\ne k_j \]

is true, then \(\Gamma_{k_1,k_2,\ldots,k_s}\) is a maximal nilpotent periodic subgroup of \(GL(n,R)\).

Lemma 3. Let \(k_1=k_2=\cdots=k_s=k\). The group \(\Gamma_{k_1,k_2,\ldots,k_s}\) is contained in some nilpotent periodic subgroup of the group \(GL(n,R)\) only when \(k=2^\alpha,\ s=2^\beta\).

From the last two lemmas it follows:

Second main theorem. A reducible nilpotent periodic group \(\Gamma_{k_1,\ldots,k_s}\), inducing in each irreducible block of degree \(n_i>0\) a maximal irreducible nilpotent periodic group determined by the number \(k_i\), where \(n_i=\varphi(k_i)\), is not only then ...

is a maximal nilpotent periodic subgroup of \(GL(n,R)\), \(n=n_1+\cdots+n_s\), when among \(k_1,k_2,\ldots,k_s\) there are at least two such numbers \(k_{j_1}\) and \(k_{j_2}\) that \(k_{j_1}=k_{j_2}=2^\alpha\).

At the same time, the following assertions follow from this.

Theorem 4. In \(GL(n,R)\), up to conjugacy in \(GL(n,R)\), there exists only a finite number of maximal nilpotent periodic subgroups.

Theorem 5. Two maximal nilpotent periodic subgroups of \(GL(n,R)\) are conjugate in \(GL(n,R)\) if their centers are conjugate in \(GL(n,R)\).

In conclusion, I express my deep gratitude to D. A. Suprunenko for discussion of a number of questions connected with the present note.

Belorussian State University
named after V. I. Lenin

Received
10 VI 1965

REFERENCES

1. D. A. Suprunenko, Solvable and Nilpotent Linear Groups, Minsk, 1958.
2. D. A. Suprunenko, Mat. sbornik, 49 (91), 3, 347 (1959).
3. D. A. Suprunenko, Mat. sbornik, 50 (92), 1, 59 (1960).
4. D. A. Suprunenko, R. F. Apatenok, Dokl. AN BSSR, 3, No. 12, 475 (1959).
5. D. A. Suprunenko, R. F. Apatenok, Dokl. AN BSSR, 5, No. 12, 535 (1961).
6. D. A. Suprunenko, R. I. Tyshkevich, Izv. AN SSSR, ser. matem., 24, 787 (1960).
7. D. A. Suprunenko, Mat. sbornik, No. 9, 3 (1961).
8. D. A. Suprunenko, R. P. Medvedeva, Dokl. AN BSSR, 2, No. 9, 363 (1958).
9. R. T. Volyachev, Izv. AN SSSR, ser. matem., 27, 1031 (1963).
10. L. Kaloujnine, Ann. Sci. École Norm. Sup., (3) 65, 236 (1948).
11. M. Hall, The Theory of Groups, Moscow, 1962.
12. L. A. Kaluzhnin, Ukr. matem. zhurn., 8, No. 3, 262 (1956); 8, No. 4, 413 (1956).
13. I. Schur, Sitzungsber. Preuss. Akad. Wiss., 619 (1911).
14. A. I. Mal’tsev, Mat. sbornik, 8 (50), 405 (1940).
15. D. A. Suprunenko, G. P. Zhavrid, Dokl. AN BSSR, 8, 320 (1958).