# Name for a certain kind of 'weakly primitive' permutation group

I have found it necessary to define the following property:

Consider a finite set $X$ and a group $H$ of permutations of $X$. Suppose for every normal subgroup $K$ of $H$ that $K$ acts faithfully on every $K$-orbit.

The property I have is that $G$ acts on $X$, not necessarily faithfully, inducing a permutation group $H$ as above.

Is there a standard name for the property possessed by $H$?

Certainly primitive permutation groups are examples for $H$, but so are fixed-point-free permutation groups for instance. On the other hand, $S_n \wr S_k$ acting on $nk$ points in the usual imprimitive way is not an example.

I think 'quasi-primitive' is already taken, and means 'every normal subgroup is transitive'. This is strictly stronger than the property I am talking about.

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If $H$ is transitive, it seems to be the case that $K$ is either faithful on each of its orbits, or is faithful on none of them,so in tht case, you seem to have just excluded the case that $K$ does not act faithfull on any orbit (which does happen, as your wreath product example shows). –  Geoff Robinson Jul 5 '11 at 18:02
Yes, that is more or less the point. Another way of saying it is that given $1 < L \unlhd K \unlhd H$ then $L$ is not contained in any point stabiliser of $H$. –  Colin Reid Jul 5 '11 at 19:18