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

D. A. Suprunenko

Sbornik. Mathematics, Tome 55 (1986) no. 1, pp. 39-54

Résumé

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.