Horrible problem about abelian groups

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Homework Help Overview

The discussion revolves around a group theory problem involving the properties of a group G, specifically focusing on the condition that for all elements a, b in G, the equation (ab)^i = (a^i)(b^i) holds for three consecutive positive integers. The goal is to demonstrate that G is abelian.

Discussion Character

  • Exploratory, Mathematical reasoning, Assumption checking

Approaches and Questions Raised

  • Participants explore the implications of the given condition, with one suggesting a reduction of the problem to the case where i = 2. Others introduce examples to illustrate the properties of the group, questioning how substitutions can lead to conclusions about the commutativity of elements.

Discussion Status

The discussion is active, with participants sharing various approaches and examples. Some have attempted to manipulate the equations to derive relationships between the elements, while others are questioning the validity of their steps and seeking clarification on the implications of their findings.

Contextual Notes

There is a focus on using specific examples and algebraic manipulation to explore the properties of the group. Participants are also engaging with the cancellation laws and how they apply to the equations presented.

Oster
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G is a group and for all elements a,b in G,
(ab)^i = (a^i)(b^i) holds for 3 consecutive positive integers.
Show that G is abelian.

I know how to prove that if (ab)^2 = (a^2)(b^2) then G is abelian. I was thinking that you could reduce the given equality integer by integer till 2 or something like that, but all I have been able to do so far is use the cancellation laws to get some gibberish. HAALP!
 
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OK, we can do this all formally, but it's best to do with an example. Let's assume that

aaaabbbb = abababab

aaaaabbbbb = ababababab

aaaaaabbbbbb = abababababab

Let's deal with the first two equalities first. Substitute abababab=aaaabbbb into aaaaabbbbb.
Can you use this to prove that abababab = babababa ??
 
baabaa black sheep
 
starting with the 2nd equation, using the left and right cancellation laws on a and b you get
aaaabbbb =babababa. Using this is with the first equation yields babababa=abababab
 
OK, now do the same with the second and the third equation.
 
3rd equation => aaaaabbbbb=bababababa
plugging into the 2nd gives bababababa=ababababab
then plug in babababa=abababab to cancel and get ab=ba
=D
thank you@!
 

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