A CAYLEY file of finite simple groups

Deficiency zero presentations for certain perfect groups

“…The presentation PI is given in [1]. The permutation generators a = (1, 2)(3, 7)(4, 8)(5, 9) 6 = (1, 2, 3)(4, 9, 5) (6,7,8) satisfy the presentation P2 and are on the Cayley file as described in [5]. Presentations P3 and P4 are obtained by putting ab =x, ab' 1 =y in PI and P2 respectively.…”

“…The presentation PI is given in [3]. Permutations a and b satisfying P2 are given on the Cayley file described in [5]. It is not hard to show that P2 may be obtained from PI by replacing b by b~l.…”

“…Notes. The presentation PI is satisfied by permutation generators a and b given on the Cayley file described in [5]. The presentation P2 is satisfied by the matrix generators…”

“…Questions concerning the efficiency of direct products are posed by Wiegold in [13]. In particular, he asks whether A 5 x A 5 is efficient and whether SL (2,5) x SL (2,5) has deficiency zero. Kenne shows that A 5 xA 5 is efficient in [8], where he also poses the question whether SL (2,5) x SL{2, 5) is efficient.…”

“…(i) SL (2,32), SL(2, 64), (ii) SL (2,27), SL(2, 49), A 7 , Sz(8), (iii) SL (2,5) x SL (2,5), SL{2, 5) x 5L (2,25). (iv) finite groups G lf G 2 , G 3 having as images SL (2,1), SL(2, 7) and SL (2,5) respectively. We also give some neat two generator, two relation presentations for certain perfect groups known to have deficiency zero.…”