TITLE: Show that every group of order 48 has a nontrivial normal subgroup.
QUESTION [7 upvotes]: Show that every group of order $48$ has a nontrivial normal subgroup.

Proof
Let $G$ denote a group of order $48$. Let $P_k$ denote the k-sylow subgroup, and $n_k$ denote the number of conjugates of $P_k$. Let $N(P_k)$ be the normalizer of $P_k$. Then,
$n_3 \equiv 1 \mod 3$ and $n_3 | 16 \implies n_3=1,4,16$
$n_2 \equiv 1 \mod 2$ and $n_2|3 \implies n_2=1,3$
Suppose that $n_3 = 4$ or $16$, and let $n_2=3$ for contradiction.
Then $n_3 = 3 \implies |N(P_3)|=16$. So we have a subgroup of order 16 in $G$.

There is a theorem in Rotman's textbook (Advanced Modern Algebra):
(Representation of Cosets). If H is a subgroup of finite index n in a group G, then there exists a homomorphism $\phi: G \rightarrow S_n$ with $\ker \phi \subseteq H$.

Using this theorem for $N(P_2)$, we know that there is a homomorphism $\phi: G \rightarrow S_3$ with kernel in $N(P_2)$. $P_3$ can only be sent to the 3-sylow subgroup of $S_3$, since the image of an element must divide the order of the pre-image. Since the 3-sylow subgroup of $S_3$ is normal, its image must also be normal. So $P_3$ is normal, a contradiction. So $n_2=1$ and we're done.
Do you think my proof is correct?
Thank you in advance

REPLY [1 votes]: I wanted to expand a bit on the specific homomorphism $\phi$ Rotman has in mind: $G$ acts on the left coset space $G/H$ by left-multiplication, which is what induces the homomorphism into $S_n$.
There are 2 things we want to be sure do not happen:
1) That the kernel is trivial. Since the order of $G$ in this case is much larger than $[G:H]!$, this does not happen ($|H| = 16$ and clearly $48 > 6$).
2) That the kernel is not all of $G$ itself (i.e., that the image is trivial). But this is ruled out by the fact that only left-multiplication by elements of $H$ will fix the coset $H$, so the kernel of $\phi$ must be contained within $H$ (and thus, in particular, NOT all of $G$).
So we know $G$ has a proper non-trivial normal subgroup (the kernel of $\phi$), which is all your question asks for (although we do not know if this subgroup is $P_2, P_3$ or $N(P_3)$).
I believe Herstein actually has a theorem that if a finite group $G$ has a subgroup $H$ such that $|G| \not \mid [G:H]!$, then $H$ contains a non-trivial normal subgroup, which is based on the same idea.
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