# Permutations, cycles

by kathrynag
Tags: cycles, permutations
 P: 603 1. The problem statement, all variables and given/known data Let t be an element of S be the cycle (1,2....k) of length k with k<=n. a) prove that if a is an element of S then ata^-1=(a(1),a(2),...,a(k)). Thus ata^-1 is a cycle of length k. b)let b be any cycle of length k. Prove there exists a permutation a an element of S such that ata^-1=b. 2. Relevant equations 3. The attempt at a solution We assume t is an element of S and a is an element S. By definition of elements of S if t is in S, we have a determined by t(1), t(2),...,t(n) Furthermore if a is in S, we have a(1), a(2)....a(n). That's as far as I get.
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P: 26,148
hi kathrynag!

just read out that equation in English …
 Quote by kathrynag ata^-1=(a(1),a(2),...,a(k))
… so what, for example, does ata^-1 do to a(2) ?
 P: 603 Does it go to 2?
 P: 603 Permutations, cycles Ok if a is defined by 1--->a(1), 2-----a(2),k--->a(k), then a^-1 is defined as a(1)-->1,a(2)--->2, a(k)--->k Then ata^-1 for a(2) is a(t(2))=a(2)
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P: 26,148
 Quote by kathrynag Does it go to 2?

a-1
sends it to 2 …

so what does ata-1 send it to?
 P: 603 a(t(2)) t(2)=2 a(2)
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P: 26,148
 Quote by kathrynag Let t be an element of S be the cycle (1,2....k)
 Quote by kathrynag a(t(2)) t(2)=2 a(2)
ah! no, you're misunderstanding the notation for a cycle …

(1,2....k) means that it sends 1 to 2, 2 to 3, … and k to 1.
so t(2) = … ?
 P: 603 3 so we are left with a(3)=4
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P: 26,148
 Quote by kathrynag 3 so we are left with a(3)=4
no!!

you're not told anything about a, are you?

a(3) is just a(3) ! :bigginr:

(and I'm off to bed … see you tomorrow!)
 P: 603 I thought I could do this: By definition of elements of S if t is in S, we have a determined by t(1), t(2),...,t(n) Furthermore if a is in S, we have a(1), a(2)....a(n).
 P: 603 So I have ata^-1=at(n) because a^-1 sends a(1)--->1,a(2)--->2,.....a(n)--->n By defininition of t, we have a(1), a(2), a(3)...a(k). We go to k because our definition of t says k<=n. For b, Let b be any cycle of length k. We have (1,2.....k). I'm not sure how to show the rest.
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P: 26,148
hi kathrynag!

(just got up …)
 Quote by kathrynag So I have ata^-1=at(n) because a^-1 sends a(1)--->1,a(2)--->2,.....a(n)--->n By defininition of t, we have a(1), a(2), a(3)...a(k).
do you mean we have (a(1), a(2), a(3),...a(k))?

anyway, before we go any further, i need to know: what exactly is S?

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