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Let $A$ be a group, where $a^2=1$ and $a$ belongs to $A$. Prove that this group is commutative.

Thank you for help.

Manos
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MaathC
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    This has to be a duplicate. – Thomas Andrews Apr 03 '14 at 05:41
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    It seems a lot of people here are assuming properties of exponents. Obviously for regular numbers, it's true $(xy)^2 = x^2y^2$, but this property isn't one of the fundamental properties of a group. Is there some theorem that says this is always the case for all groups? – Jared Apr 03 '14 at 05:56
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    @Jared: $(xy)^2 = x^2y^2$ is not the case for all groups. For $(xy)^2 = x^2y^2$ then $xyxy = x^2y^2$ implying $xy = yx$, so $(xy)^2 = x^2y^2$ if and only if the group is abelian. – Robert Lewis Apr 03 '14 at 06:05
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    @RobertLewis I didn't think so (was specifically thinking about invertible matrices). I thought your proof used this as well, but now I see $(xy)^{-1} = y^{-1}x^{-1}$ is certainly correct since $x(yy^{-1})x^{-1} = 1$. – Jared Apr 03 '14 at 06:11
  • You need to be more precise. Does $a^2=1$ stand for every $a\in A$ ? – Sylvain Biehler Apr 03 '14 at 09:21
  • $ab = a1b = a(ab)^2b = aababb = a^2bab^2 = 1ba1 = ba$. – Magdiragdag Apr 03 '14 at 19:32

5 Answers5

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Let $a,b$ be elements of the group. We want to show that $a * b = b * a$. But this is equivalent to $a * b * a = b$ which is equivalent to $(a * b) * (a * b) = 1$, which is true by our hypothesis that $x * x = 1 $ for every $x$ in the group.

Manos
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  • I see it now, it would have been easier though (for me) if you had written $(aba)b = (b)b$. – Jared Apr 03 '14 at 05:49
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We have: $x^2 = e$, and $y^2 = e$. So: $(xy)^2 = e = e*e = x^2y^2$. So $(xy)^2 = x^2y^2$. Thus: $xyxy = xxyy$. So: $x^{-1}xyxyy^{-1} = x^{-1}xxyyy^{-1}$. Since $x^{-1}x = yy^{-1} = e$, and $ex = x$, $ey = y$, you have : $yx = xy$ for all $x, y$ proving the group commutative.

DeepSea
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  • How can you claim $(xy)^2 = (xy)(xy) = xxyy$ without assuming commutativity? – Jared Apr 03 '14 at 05:52
  • OK, I think I understand now. $(xy)(xy) = 1$, which must also equal $(xx)(yy) = 11 = 1$. But it seems like it's easier to simply state $xyxy = x(yx)y$ and if that equals $xxyy$ (which it does since they are both the identity) then it must be valid to swap the middle term, i.e. $x * (y * x)* y = x * (xy)y$ therefore $yx = xy$. – Jared Apr 03 '14 at 06:06
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Here's my take on it:

Since $a^2 = 1$ for all $a \in G$, $a = a^{-1}$ for all $a \in G$. Let $x, y \in G$; then $xy = (xy)^{-1} = y^{-1}x^{-1} = yx$, since $x = x^{-1}$, $y = y^{-1}$, and $xy = (xy)^{-1}$. $G$ is indeed abelian. QED.

Hope this helps. Cheerio,

and as always,

Fiat Lux!!!

Robert Lewis
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It's not true. Choose for example: $$ a_1=\pmatrix{0 & 1\\1&0} \text{ and } a_2=\pmatrix{1 &0\\0&-1}. $$ For both you have $a_k^2=1$, but the generated group is not commutative.

draks ...
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    Did he say anything about generators? $(a_1a_2)^2\neq 1$ so your group doesn't qualify. – Thomas Andrews Apr 03 '14 at 05:36
  • @ThomasAndrews no, I just thought I construct a group as a counterexample. It just happened that my example also generates a group. Is my reasoning wrong?...oh I see – draks ... Apr 03 '14 at 05:38
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    Well, the question is a mess, but almost every algebra book in existence asks a question: If every element of a group, when squared, is the identity, then that group is commutative. That's how every other person here read the question. – Thomas Andrews Apr 03 '14 at 05:40
  • @ThomasAndrews looks like... – draks ... Apr 03 '14 at 05:41
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I think this is pretty straigtforward. All you need to use is associativity. We wish to prove $a*b = b*a$. First multiply both sides by $a$:

$$ a*(a*b) = a*(b*a) \\ (a*a)*b = (a*b)*a \\ b = a*b*a $$

Now, multiply both sides by $b$: $$ b * (a*b) = b*(b*a) \\ b*a*b = (b*b)*a \\ b*a*b = a $$

Now you can plug in $a = b*a*b$ into $b = a*b*a$:

$$ b = (b*a*b) * b * a \\ b = b*a*(b*b)*a\\ b = b*(a*a) \\ b=b \text{, q.e.d.} $$

edit as per the comment, here is my edited proof:

I can prove it through the following: \begin{align} b =& b*(a*a) \\ b =& b *a* (b*b) *a \text{, inserted } 1 \text{, between the } a\text{'s} \\ b =& (b*a*b)*b*a \\ b*(b*a) =& (b*a*b)*b*a * b*a\\ (b*b)*a =& (b*a*b)*\left((b*a)*(b*a)\right)\text{, since } (b*a)*(b*a) = 1 \\ a =& b*a*b \\ a*b =& (b*a*b)*b = b*a \end{align}

Jared
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    This isn't a proof. You've started out assuming the thing you wanted to show, and then deduced something that's also true from that (possibly false assumption). One can 'prove' many things by assuming something that needn't be true, so in a mathematical proof you should only start with what you know is true (i.e., not the statement you wish to prove), and then deduce the thing you want to prove. Your argument would probably work backwards however, since $b=b$ is certainly true... – ah11950 Apr 03 '14 at 07:14
  • I'm not totally sure you're right, but I definitely see your logic. I guess I proved $\left(ab = ba \wedge aa=1\wedge bb=1 \right)\rightarrow b = b$ (and I can certainly see how that's a trivial thing, since $\text{anything} \rightarrow true$ is a tautology). – Jared Apr 03 '14 at 07:24