1. The problem statement, all variables and given/known data Show that any group of order 35^3 has a normal subgroup of order 125. 2. Relevant equations 3. The attempt at a solution Is this a valid proof? Let G be an arbitrary group of order 353. Note that 353 = 5373. Thus, by Sylow's first theorem, there is a sylow p-subgroup of order 125, which we refer to as H. But then, by Sylow's second theorem, it follows that H is conjugate to itself in G. Hence, H is normal in G.