linguistic:paraphrasing, quality:corruption
Within a group

Every finite group whose order is divisible by p contains a subgroup which is a non-trivial p -group, namely a cyclic group of order p generated by an element of order p obtained from Cauchy's theorem. In fact, it contains a p -group of maximal possible order: if | G | = n = p k m {\displaystyle |G|=n=p^{k}m} where p does not divide m, then G has a subgroup P of order p k, {\displaystyle p^{k},} called a Sylow p -subgroup. This subgroup need not be unique, but any subgroups of this order are conjugate, and any p -subgroup of G is contained in a Sylow p -subgroup. This and other properties are proved in the Sylow theorems.