Ranks, Subdegrees and Suborbital Graphs of Direct Product of the Symmetric Group Acting on the Cartesian Product of Three Sets

Abstract

Transitivity and Primitivity of the action of the direct product of the symmetric group on Cartesian product of three sets are investigated in this paper. We prove that this action is both transitive and imprimitive for all 2 n ≥ . In addition, we establish that the rank associated with the action is a constant 3 2 . Further; we calculate the subdegrees associated with the action and arrange them according to their increasing magnitude.

Transitivity Action of the Cartesian Product of the Alternating Group Acting on a Cartesian Product of Ordered Sets of Triples

“…gap> Arrangements([8,9,10,11,12,13,14],3); 𝑆 [3] = { [ 8,9,10 ], [ 8,9,11 ], [ 8,9,12 ], [ 8,9,13 ], [ 8,9,14 ], [ 8,10,9 ], [ 8,10,11 ], [ 8,10,12 ], [ 8,10,13 ], [ 8,10,14 ], [ 8,11,9 ], [ 8,11,10 ], [ 8,11,12 ], [ 8,11,13 ], [ 8,11,14 ], [ 8,12,9 ], [ 8,12,10 ], [ 8,12,11 ],…”

“…The ranks and subdegrees associated with this action for 𝑛 ≥ 4 is 8 ; and 1, (n − 1), (n − 1) 2 , (n − 1) 3 respectively. Muriuki et al [10] showed that for the action of direct product of three symmetric groups on Cartesian product of three sets, the action is both transitive and imprimitive for all 𝑛 ≥ 2 and the associated rank is 2 3 . Mutua et al [11] showed that the direct product of 𝑆 𝑛 × 𝐴 𝑛 on 𝑋 × 𝑌 has its action both transitive and imprimitive when 𝑛 ≥ 3.…”

On Suborbits and Graphs Associated with Action of Alternating Groups on Cartesian Product of Two Sets

“…Gikunju [17] constructed and investigated the suborbital graphs corresponding to the action of direct products of symmetric groups Sn on a Cartesian product of three sets. It was showed that the suborbital graphs Gi, i = 1, 2, • • • , n corresponding to the non-trivial suborbits Oi, i = 1, 2, • • • , 6 are disconnected but G7 is connected each with girth 3 for all n > 2.…”