Sylow's Theorem Proof for Group of Order 35^3 | Validity Check

[Syrus]

Homework Statement

Show that any group of order 35^3 has a normal subgroup of order 125.

Homework Equations

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.

 

[Chaos2009]

I believe you are on the right track, however, I believe you have made an assumption. Sylow's second theorem says that all the Sylow 5-subgroups are conjugate to each other. This does imply normality of H iff H is the only Sylow 5-subgroup of G. I think you can use Sylow's third theorem to show that H is the only Sylow 5-subgroup.