quasiprimitive

In abstract algebra, a group G is called quasiprimitive if and only if its soluble normal subgroups are trivial. This concept is related to the study of primitive groups, which are groups acting faithfully on a set such that no proper non-trivial subgroup preserves any non-trivial partition of the set. A quasiprimitive group is a generalization of a primitive group.

A key property of quasiprimitive groups is that they are closely linked to primitive groups. Specifically, a group G is quasiprimitive if and only if it is primitive or G is a direct product of primitive groups and G acts faithfully on the direct product of the sets on which these primitive groups act. More precisely, if G is a group and X is a set on which G acts, then G is quasiprimitive on X if its soluble normal subgroups are trivial and G acts faithfully on X.

The study of quasiprimitive groups is important in understanding the structure of general groups. Their properties allow for a finer classification of group actions and have implications in areas such as permutation group theory and the study of highly symmetric structures. For instance, if a group G acts transitively on a set X, and G is quasiprimitive on X, then G possesses a strong structure related to primitive groups.