Role of sylow's theorem on permutation group and automorphic character

Abstract

In the permutation group. Every group is isomorphic to a permutation group. In permutation group the smallest subgroup of Sn containing (12) and (1, 2, 3, , n) is Sn. The converse of lagranges theorem does not hold, in general, as O(An) is divisible by 6 but it has no sub group of order 6. It is also notable that quotient group of cyclic (abilian) groups are cyclic (abelian) but converse do not hold.