Geodesic
Generating sets of conjugate involutions of the groups $SL_n(q)$ for $n=4,5,7,8$ and odd $q$
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 1, pp. 62-69 Cet article a éte moissonné depuis la source Math-Net.Ru

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In 2009 J. M. Ward answered for sporadic and alternating groups and for projective special linear groups $PSL_n(q)$ over a field of odd order $q$ except for the case $q=9$ for $n\geq 4$ and, for $n=6$, the case $q\equiv 3\mod 4$ Question 14.69c from The Kourovka Notebook posed by the second author of the present paper: For every finite simple nonabelian group $G$, find the minimum number $n_c(G)$ of generating conjugate involutions whose product is $1$. It is known that $n_c(G)\geq 5$ for any simple nonabelian group $G$. We discard the constraint $q\neq 9$ for the dimensions $n=4,5,7,8$. It turns out that in these dimensions the generating quintiples of conjugate involutions with the product equal to 1 for special linear groups $SL_n(q)$ and, consequently, for $PSL_n(q)$, specified by Ward, are also suitable for $q=9$.
Keywords: spacial linear group over a finite field, generating triples of conjugate involutions. 
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I. Yu. Efimov; Ya. N. Nuzhin. Generating sets of conjugate involutions of the groups $SL_n(q)$ for $n=4,5,7,8$ and odd $q$. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 27 (2021) no. 1, pp. 62-69. http://geodesic.mathdoc.fr/item/TIMM_2021_27_1_a5/

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