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Abstract

The problem of finding a necessary and sufficient condition for the triviality of a (G,x)-space leads us to study and classify the properties of a minimal (k)-group G acting on N = {1,2,...,N} (i.e. for each partition P = {X₁,X₂,...X[sub k]} on N , there exists 1 ≠ g ε G with g(X[sub i]) = X[sub i] for all i ) . In this thesis, we construct and classify all the minimal (k)-groups of degree ≤ 3k and tabulate the results. We also obtain all the non-primitive (k)-groups of degree ≤ 4k . We then apply our results to determine., which of the (G, X) -spaces are trivial. We found that for some of the (k)-groups, no appropriate character exists while for most of the remaining, the associated character has range {1,-1} . Finally, a table has been made to show the number of the appropriate characters on some (k)-groups.