IJPAM: Volume 106, No. 1 (2016)

PROPERTIES AND INVARIANTS ASSOCIATED WITH
THE ACTION OF THE ALTERNATING GROUP
ON UNORDERED SUBSETS

R. Gachimu$^1$, I. Kamuti$^2$, L. Nyaga$^3$, J. Rimberia$^4$, P. Kamaku$^5$
$^{1,3,5}$Pure and Applied Mathematics Department
Jomo Kenyatta University of Agriculture and Technology
P.O. Box 62000-00200, Nairobi, KENYA
$^{2,4}$Mathematics Department
Kenyatta University
P.O. Box 43844-00100, Nairobi, KENYA


Abstract. The transitivity, primitivity, rank and subdegrees, as well as pairing of the suborbits associated with the action of the alternating group $A_n$, on unordered $r-$element subsets of a set $X=\{1,2,\cdots,n\}$ of $n$ letters, have not received any attention. In this paper, we prove that this action is transitive. We also show that the action is imprimitive if and only if $n=2r$. In addition, we establish that the rank associated with the action is a constant $r+1$ if and only if $n\geq2r$, except for $r=2$ in which case the rank is 4 if $n=4$, but is 3 for all $n\geq5$. Further, we calculate the subdegrees associated with the action and arrange them according to their increasing magnitudes. Finally, we show that all the suborbits of the action, with the exception of some non-trivial suborbits corresponding to the actions of $A_3$ and $A_4$ on the set of unordered pairs, are self-paired.

Received: September 21, 2015

AMS Subject Classification: 05E18

Key Words and Phrases: alternating group, action, rank, subdegrees, unordered $r-$element subset, suborbit

Download paper from here.



DOI: 10.12732/ijpam.v106i1.27 How to cite this paper?

Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2016
Volume: 106
Issue: 1
Pages: 333 - 346


Google Scholar; DOI (International DOI Foundation); WorldCAT.

CC BY This work is licensed under the Creative Commons Attribution International License (CC BY).