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I need to prove the following fact: if $G$ is a non abelian group of order $6$, then $G$ is isomorphic to the dihedral group $D_3$. Here what I have done:

Suppose that $G$ is non abelian group of order $6$. Since $G$ is not abelian, it cannot have a element of order $6$. (This is correct, since if $G$ had a element of order $6$, then $G$ would be cyclic, and hence abelian, a contradiction). Also, it cannot have elements of order 1, by the same argument as before.

But I got stucked in here. I'm struggling with the two other possibilities of the order of the elements in $G$: order $2$ and $3$. Any help or hints that you can bring me would be appreciated.

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If you're allowed to use Cauchy's theorem, you may argue as follows:

Since $2$ and $3$ are prime numbers that divide $6$, we know that there are elements $a$ and $b$ of orders $2$ and $3$ respectively. Therefore, $\{e,a,b,b^2,ab,ab^2\} \subseteq G$. Since $G$ has $6$ elements, this subset is all of $G$.

Now the only thing left to do is to see how $a$ and $b$ interact with each other. Remember that $a=a^{-1}$ and $b^2=b^{-1}$. Now consider $aba$:

$aba=e \implies b=e$. Contradiction

$aba=a \implies b=a$. Contradiction

$aba=b \implies ab=ba$. But $G$ is assumed not to be Abelian. Contradiction.

$aba=ab \implies a=e$. Contradiction

$aba=ab^2 \implies a=b$. Contradiction

Hence, $aba=b^2$. This means that $aba=b^{-1}$. Hence, $G$ is isomorphic to $D_3$.

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  • $\begingroup$ I have a question: where are the elements of the dihedral group $D_3$ to establish the isomorphism$? The thing is that we haven't talked about the Cauchy Theorem yet so I'm not so sure if the professor wants me to use such theorem. I really want to understand what's going on in your solution! I would really appreciate that you explain it to me in detail. $\endgroup$ – user573497 Sep 9 '18 at 13:51
  • $\begingroup$ @user573497 Well, it actually depends on how you define $D_n$. Generally, the way $D_n$ is defined in most standard textbooks is that $D_n$ is generated by two elements $a$ and $b$ such that $a^2=1$,$b^n=1$ and $aba=b^{-1}$. Do you have a different definition for $D_n$ or $D_3$ in particular? If yes, what is it? $\endgroup$ – stressed out Sep 9 '18 at 13:56
  • $\begingroup$ @user573497 I'm not sure if you can completely avoid Cauchy's theorem or not, but you can use Lagrange's theorem to conclude that the only possible orders for the elements of $G$ are $1,2,3,6$. You can exclude $1$ and $6$ and conclude that an element of $G$ that is not identity must have order $2$ or $3$. I believe if you're lucky, this should work because the order of $G$ is very small. In fact, I just checked that this idea works. Can you finish this argument on your own? Let me know if you needed help. $\endgroup$ – stressed out Sep 9 '18 at 13:59
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So you know that there is a subgroup of order 2 in $G$, call it $H$. We can define a map $G \to S_3$ (or $D_3$ but here it's more useful to think of it as $S_3$) by looking at group actions on the different cosets of $H$. Anything that fixes all cosets must be in $H$. Now kernels of maps are always normal but I claim that $H$ is not normal. Can you prove this step? This implies that the kernel cannot be all of $H$, i.e it must be just the identity. Now we have an injective homomorphismm between two finite groups, so they must be isomorphic.

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