On a family of groups defined by Said Sidki

Abstract

In a paper in 1982, Said Sidki defined a 2-parameter family of finitely presented groups Y(m,n) that generalise the Carmichael presentation for a finite alternating group satisfied by its generating 3-cycles (1,2,t) for t⩾3. For m⩾2 and n⩾2, the group Y(m,n) is the abstract group generated by elements a1,a2,⋯,am subject to the defining relations ain=1 for 1⩽i⩽m and (aikajk)2=1 for 1⩽i<j⩽m and 1⩽k⩽[n2]. Sidki investigated the structure of various subfamilies of these groups, for small values of m or n, and has conjectured that they are all finite. Sidki's conjecture remains open. In this paper it is shown that for all m⩾3, the group Y(m,6) is finite, and is isomorphic to a semi-direct product of an elementary abelian 2-group of order 2m(m+3)/2 by Y(m,3)≅Am+2. Also we exploit a computation for the group Y(3,8) to prove that Y(m,8) is a finite 2-group, for all m.