Let G be a finite p-group
and G a minimal faithful permutation representation of G possessing the
minimal number of generators of the centre of G transitive constituents.
One surmises that the induced representation, G′, of the centre of G, is
minimal. The conjecture is validated subject to either of the hypotheses
|G|≦ pf except G = Q8× Z1 or Z(G)≅n copies of the cyclic group of order pm
and is trivial when G is abelian. However, a group of order p6 shows the
conjecture to be false for p odd, also. The converse problem of extending minimal
representations of Z(G) to minimal representations of G is also, in general, not
possible.
