Classical Groups of Large Rank as Hurwitz Groups

Hurwitz Groups of Intermediate Rank

“…If v = a 3 , then vψ(X) = a 4 . Therefore, (11) follows from (6). Since ψ(XY ) has order 7, we have a 4 = a 3 ψ(X) = a 3 , which -together with (6) -implies that inclusion (10) holds in this case, too.…”

“…The technique is due to A. Lucchini, M. C. Tamburini and J. S. Wilson; see, for example, [6,7]. Most of the statements in this section may be regarded as non-trivial refinements of similar results in [6] and [7].…”

“…Remark 2.8. A special case of Lemma 2.6, namely when b 1 = tc 1 and b 2 = tc 2 for some handle (c 1 , c 2 ) with c 1 , c 2 ∈ and for some t ∈ R, appears in [2,6,7].…”

“…We just mention a recent survey [10], where an overview of the known results is given. Particular attention is paid to the case of classical groups over various rings, especially over finite fields or the ring Z of integers; see [3,6,7]. As is shown in [3], many linear classical groups of rank less than 18 are not Hurwitz.…”

“…As is shown in [3], many linear classical groups of rank less than 18 are not Hurwitz. On the other hand, for all sufficiently large ranks, the groups SL n (q), Sp 2n (q), SU 2n (q) and + 2n (q) for any prime power q, and SU 2n+1 (q) and 2n+1 (q) for any odd prime power q, are known to be Hurwitz; see [6,7]. For example, Lucchini, Tamburini and Wilson proved the following theorem [7,Corollary 1].…”