Given α◦β=(1532)(14)(35)
How do we get from the given to = ( 1 4 5 2 ) ( 3 ) = ( 1 4 5 2 ) = (4 1 3 5 2) ?
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Work out where each number maps to. – Tim Seguine Apr 04 '14 at 10:23
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1$(1 4 5 2)(3)=(1 4 5 2)$?. Will someone explain. I don't understand this notation. – Guy Apr 04 '14 at 10:25
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It's cyclic notation, the $(3)$ means keep $3$ constant, so it can be omitted. – George1811 Apr 04 '14 at 10:35
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It is easiest to write this out and compose from right to left as follows $$ \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 5 & 1 & 2 & 4 & 3 \end{pmatrix}\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 2 & 3 & 1 & 5 \end{pmatrix}\begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 1 & 2 & 5 & 4 & 3 \end{pmatrix} $$ $$ = \begin{pmatrix} 1 & 2 & 3 & 4 & 5 \\ 4 & 1 & 3 & 5 & 2 \end{pmatrix} = (1452) $$ For instance, you check $1 \rightarrow 1 \rightarrow 4 \rightarrow 4$
Prahlad Vaidyanathan
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That is very clear! Thanks. What about if we start counting from the 5th term in the pre last bracket, can we also say that it all equals to, say, (5 2 1 4), instead of (1 4 5 2)? – user138913 Apr 04 '14 at 10:42
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@user138913 as long as the order of the numbers in the cycle remains the same, they represent the same cycle. – Tim Seguine Apr 04 '14 at 11:29