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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?

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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 $G \cong A_4.$

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  • $\begingroup$ What is the homomorphism literally? Is it conjugation? $\endgroup$ – Meitar Jan 3 '15 at 17:06
  • $\begingroup$ 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. $\endgroup$ – Krish Jan 3 '15 at 17:12
  • $\begingroup$ Oh I get it. Sorry I am so frustrated by another question and it exhausted me and my brain. Thank you! :) $\endgroup$ – Meitar Jan 3 '15 at 17:14
  • $\begingroup$ How can I say unequivocally that K={e}? $\endgroup$ – Meitar Jan 3 '15 at 17:22
  • $\begingroup$ $P$ 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 $\{e\}.$ $\endgroup$ – Krish Jan 3 '15 at 17:27

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