Prove that a group of order 36 must have a normal subgroup of order 3 or 9.
Let n2 be the number of 2-Sylow subgroups of G (with |G|=36). Then
n must be 1 or 3.
Let n3 be the number of 3-Sylow subgroups of G. then n3=1 or n3=4
if n3=1 we have 1 3-sylow group of order 9. and it is also a normal group(from sylow theorem )
if n2=1 there is normal group of order 4 but I cant show normal group of order 3.
Best Answer
Assume that $G$ has $4$ Sylow $3$-groups, as you noted $n_3=(1+3k)|4, n_3\neq1$. Defining the conjugation action on the set of these $4$ Sylow $3$-groups, we have the induced homomorphism $\phi: G\longrightarrow S_4$. So $\frac{G}{\ker\phi}\hookrightarrow S_4$, but $3^2$ divides |G|=36 and does not divide $|S_4| = 24$, so $\ker\phi\neq 1$ and of course $\ker\phi\neq G$. This means that there must be non-trivial kernel, which is a non-trivial normal subgroup of G.