The Ganita-Sâra-Sangraha of Mahâvïrâcârya with English

subgroups cannot involve two operators whose orders are powers of prime numbers and whose product has an order which is a power of another prime number. In particular, a solvable group cannot involve two such operators. The second question to which we referred above is as follows: Is there a group which is not the product of some one of its possible sets of non-conjugate Sylow subgroups? I t is well known that a necessary and sufficient condition that a group is the direct product of its Sylow subgroups is that we arrive at identity by forming the successive groups of inner isomorphisms, but no general criterion as regards whether a group is a product of a set of non-conjugate Sylow subgroups seems to have been found.