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 Mat. Sb. (N.S.), 1985, Volume 127(169), Number 1(5), Pages 40–54 (Mi msb1956)

Subspaces generated by the rows of circulants, and minimal irreducible linear groups

D. A. Suprunenko

Abstract: The author describes the soluble minimal irreducible subgroups of $GL(pq,K)$, where $p$ and $q$ are prime numbers, $p>q$, $q\nmid p-1$, and $K$ is an arbitrary subfield of the field of real numbers. He proves that up to conjugacy, there exist exactly 4 soluble minimal irreducible subgroups in $GL(pq,K)$: $G_1=D_1H_1$, $G_2=D_2H_1$, $G_3=D_3H_2$, and $G_4 = D_4H_3$, where each $D_i$ is a Sylow 2-subgroup of $G_i$ and $H_1$, $H_2$, and $H_3$ are minimal transitive groups of permutation matrices of degree $pq$, $G_1$ and $G_2$ are metabelian groups, each of which is generated by two matrices, and $G_3$ and $G_4$ are soluble groups of class 3 with three generators:
$$|G_1|=2^{m_{pq}}pq, \quad |G_2|=2^{m_p+m_q}pq, \quad |G_3|=2^{qm_p}p^mq, \quad |G_4|=2^{pm_q}pq^l,$$
where $m_d$ is the order of the number 2 modul $d$, $m$ is the order of $p$ modulo $q$, and $l$ is the order of $q$ modulo $p$.
The properties of subspaces generated by the rows of circulants over a prime finite field are investigated. The connection between these properties and the problem of describing certain classes of minimal irreducible linear groups is indicated.
Bibliography: 18 titles.

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English version:
Mathematics of the USSR-Sbornik, 1986, 55:1, 39–54

Bibliographic databases:

UDC: 512.5
MSC: Primary 20G20; Secondary 20F16

Citation: D. A. Suprunenko, “Subspaces generated by the rows of circulants, and minimal irreducible linear groups”, Mat. Sb. (N.S.), 127(169):1(5) (1985), 40–54; Math. USSR-Sb., 55:1 (1986), 39–54

Citation in format AMSBIB
\Bibitem{Sup85} \by D.~A.~Suprunenko \paper Subspaces generated by the rows of circulants, and minimal irreducible linear groups \jour Mat. Sb. (N.S.) \yr 1985 \vol 127(169) \issue 1(5) \pages 40--54 \mathnet{http://mi.mathnet.ru/msb1956} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=791316} \zmath{https://zbmath.org/?q=an:0596.20040|0575.20043} \transl \jour Math. USSR-Sb. \yr 1986 \vol 55 \issue 1 \pages 39--54 \crossref{https://doi.org/10.1070/SM1986v055n01ABEH002990} 

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This publication is cited in the following articles:
1. Platonov V.P., “Minimal irreducible linear groups and representation of finite groups”, Dokl. Math., 79:1 (2009), 125–127
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