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On a family of groups defined by Said Sidki
Author(s) -
Conder Marston D. E.
Publication year - 2018
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms.12106
Subject(s) - mathematics , combinatorics , abelian group , group (periodic table) , conjecture , finite group , order (exchange) , cyclic group , product (mathematics) , direct product , alternating group , discrete mathematics , symmetric group , geometry , physics , finance , quantum mechanics , economics
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 elementsa 1 , a 2 , ⋯ , a msubject to the defining relationsa in = 1 for 1 ⩽ i ⩽ m and( a ika jk ) 2 = 1 for 1 ⩽ i < j ⩽ m and 1 ⩽ k ⩽ [ n 2 ] . 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 2 m ( m + 3 ) / 2by Y ( m , 3 ) ≅ A m + 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 .