(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

This is a problem from a chapter entitled "Permutation Groups" of an abstract algebra text.

1. Let α = ( 1 3 5 7 ) and β = (2 4 8) o (1 3 6) ∈ S_{8}Find α o β o α^{-1}.

2. Let α = ( 1 3) o (5 8) and β = (2 3 6 7) ∈ S_{8}Find α o β o α^{-1}.

2. Relevant equations

S_{n}is the set of all permutations on I_{n}, where I_{n}={1,2,3,...,n}

Also, o is known to be associative, but not commutative.

α and β are conjugate if there exists γ ∈ S_{n}such that γ o α o γ^{-1}= β

Then, let π = (i_{1}i_{2}... i_{l}) ∈ S_{n}be a cycle. Then for all α ∈ S_{n}, α o π o α^{-1 = (α(i1) α(i2) ... α(il)) 3. The attempt at a solution I was able to calculate other problems easy enough that did not contain the composition of permutation cycles. Also, I can write the composition as a two row notation instead of a cycle, but then I don't know which elements I use when I calculate the conjugate against alpha. If I left the beta as a composition, maybe I could use the associative property and apply one element first, but I'm at a loss. A worked out solution would be really great- my professor assigned me 11 of these calculations, saying that they would be "really easy" so this many will not be a big deal. Thanks, prof.}

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# Permutation Groups

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