@Jim It seems as if somebody deleted your question. So let us hope that we both remember well enough what we wrote. For the permutation group itself, no element of the group is redundant. Talking about redundant elements seems to make most sense for a generating set of the permutation group. In that case, the size of the generating set of Yi will be the sum of the sizes of the generating sets of the two involved permutation groups. Still seems greater than T to me.