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I want to make sure the following equality is true:

$(1,2,3)\circ (3,4,5)= (1, 2, 4, 5, 3)$

I don't know how to check if that's true.

Thank You

N. F. Taussig
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Sylvester Stallone
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1 Answers1

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You are close, but off bit a little bit. We see that in the product $(123)(345)$, we can check what the product is by considering each possible input, then following it through the function composition,

  • $1\to1\to2$
  • $2\to2\to3$
  • $3\to4\to4$
  • $4\to5\to5$
  • $5\to3\to1$

Therefore, we have that $(123)(345) = (123)\circ(345) = (12345)$.

Edit: @Arthur reminded me that some use algebraic notation instead of function notation regarding multiplication of permutations. In the above computation, I assumed that $(123)(345)(x) = (123)[(345)(x)]$. Using the algebraic convention, we have that

  • $1\to2\to2$
  • $2\to3\to4$
  • $4\to4\to5$
  • $5\to5\to3$
  • $3\to1\to1$

So $(123)(345) = (345)\circ(123)=(12453)$.

Santana Afton
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    This depends on whether the permutations are applied right-to-left or left-to-right. If they're applied right-to-left, then this is the correct answer. If they're applied left-to-right, then the result in the original question is true. I usually apply them right-to-left, but since that seems to be the exact thing differentiating your answer form the OP's answer, I feel it deserves a mention. – Arthur Apr 08 '17 at 13:03
  • Good point! I assumed that since Sylvester used the $\circ$ notation he was following right-to-left convention, but I'll edit my answer to address that. Thanks! – Santana Afton Apr 08 '17 at 13:04