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Let $G$ be a group of order $12.$ Prove that either $G$ has a normal $ 3$-Sylow subgroup or $ G$ is isomorphic to $A_4$.

I know that $|G|=12=2^23$ and that either $n_3=1$ and there is a $3$-sylow subgroup, or either $n_3=4$. If $n_3=4$ then there are $4$ $3$-sylow subgroups whose intersection is $\{e\}.$ That means there are $8$ elements of order $3,$ leaving $4$ element of order that is not $3$. There is a $2$-sylow subgroup of order $4.$ Its elements are of order dividing $4.$ This subgroup is normal. There is the identity, $8$ elements of order $3$ and $3$ of order $2$ or $4.$ So there is one element of order $6$, I guess. But I don't know how to continue and show isomorphism. Any help?

Meitar
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1 Answers1

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$G$ acts by conjugation on the 3-Sylow subgroups. So we have a homomorphism $\phi: G \rightarrow S_4.$ Let $K$ = ker$\phi.$ Then $K \leq N_G(P),$ where $P$ is a 3-Sylow subgroup of $G$. From this conclude that $K = \{e\}.$ So the map $\phi$ is injective. Now $G$ contains eight element of order 3. The number of elements of order 3 in $S_4$ is also eight and all are contained in $A_4.$ This shows that $G \cong A_4.$

Krish
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  • What is the homomorphism literally? Is it conjugation? – Meitar Jan 03 '15 at 17:06
  • whenever a group $G$ acts on a set $X,$ this will induce a group homomorphism $G \rightarrow S_X, g \mapsto \sigma_g$ where $\sigma_g : X \rightarrow X$ is given by $x \mapsto g \cdot x.$ this a general fact, follows from the definition of group action. – Krish Jan 03 '15 at 17:12
  • Oh I get it. Sorry I am so frustrated by another question and it exhausted me and my brain. Thank you! :) – Meitar Jan 03 '15 at 17:14
  • How can I say unequivocally that K={e}? – Meitar Jan 03 '15 at 17:22
  • $P$ is not normal by assumption. so $N_G(P) = P.$ and $K$ is the largest normal subgroup of $G$ that is contained in $P$ (this is also a general fact). this forces that $K$ has to be ${e}.$ – Krish Jan 03 '15 at 17:27
  • How can you tell that NG(P)=P if P is not normal? It's a bit confusing... What fact did you use in addition? – Meitar Jan 03 '15 at 17:38
  • From Sylow's theorem, $[G : N_G(P)] = n_3 = 4.$ So $|N_G(P)| = 3$ and $P \leq N_G(P).$ – Krish Jan 03 '15 at 17:46
  • Is the stabilizer of P also the normalizer? To be honest, when my teacher taught it, they never referred the normalizer nor its index :( – Meitar Jan 03 '15 at 18:20
  • yes. for conjugate action, stabilizer is equal to the normalizer. for the index you an look at any algebra book. – Krish Jan 03 '15 at 18:43
  • Let us continue this discussion in chat. – Meitar Jan 03 '15 at 19:27