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Is the mathematics implied by this joke accurate? As in, is the mathematical portion of the pun a correct mathematical statement?

Q: Why do motorcycle gang members use their motorcycles to get to work?

A: Because members of cyclical groups commute.

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closed as not a real question by Sasha, lhf, J. M., Asaf Karagila, Thomas Andrews Dec 22 '11 at 15:42

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.

    
Please define "work". –  Dustan Levenstein Dec 22 '11 at 15:29
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I voted to close as not a real question. –  Sasha Dec 22 '11 at 15:32
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I found the answer exceedingly helpful, but since it seems people think the question isn't a good fit for the site, I'll happily delete it as soon as the interface allows me to do so (which it says I cannot for two days). –  Daniel Dec 22 '11 at 16:14
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I voted to re-open this question because after the edit I think it's a reasonable one and I agree with lumbric's assessment. Hopefully reopening it will prevent its deletion (consider this to be my vote against deletion, so the next 10k+-user who wishes to delete the thread should leave a comment instead of voting to delete). –  t.b. Jan 3 '12 at 9:54
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I think the joke could be tightened a bit to "Why do Hell's Angels never work from home? Because members of a cyclical group always commute." –  Peter Taylor Jan 3 '12 at 11:02
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1 Answer

up vote 7 down vote accepted

Yes, cyclic groups are Abelian (their elements commute).

Proof:

Let be $g$ the generator for group $G$, that is $G = \{g^n|n\in \mathbb{N}\}$. For $a,b \in G$:

$a \cdot b = g^k \cdot g^i=g^{k+i}=g^{i+k}=g^i\cdot g^k=b\cdot a$

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The answer to the joke would be better worded as: "Because members of a cyclical group commute." The original phrasing is ambiguous, and could be reasonably but nonsensically interpreted as referring to multiplying elements from two different groups. –  Michael Joyce Dec 22 '11 at 15:44
    
Thanks, @MichaelJoyce and lumbric. Very helpful. –  Daniel Dec 22 '11 at 16:15
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