Why does this equality hold? $\begin{array}{rl} \langle a,c| &ca^{-1}cac^{-1}aca^{-1}c^{-1}ac^{-1}a^{-1}ca^{-1}c^{-1}a,\\ &ac^{-1}aca^{-1}cac^{-1}a^{-1}ca^{-1}c^{-1}ac^{-1}a^{-1}c\rangle\\ =\langle a,c| &aca^{-1}cac^{-1}aca^{-1}c^{-1}ac^{-1}a^{-1}ca^{-1}c^{-1}\rangle\\ \end{array}$ I'm asking this cause I don't understand the last line of this calculation:
concrete equality of group presentations
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group-theory
group-presentation
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3If I see correctly, the relation in the second group is simply obtained from the first relation in the first group by conjugating with $a$. So the question is: what is the connection between the first and the second relation in the first group. – 2011-06-09
1 Answers
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As Theo remarked in the comments, the relation in the second group is obtained from the first relation in the first group by conjugating with $a$.
The second relation in the first group is obtained from the first relation in the first group by inverting it and then conjugating with $c^{-1}a$.
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0uh, you're right. this was non-trivial... thank you – 2011-06-09