My Math Forum Question on Group of Permutations

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 November 6th, 2010, 07:05 PM #1 Newbie   Joined: Sep 2010 Posts: 22 Thanks: 0 Question on Group of Permutations Question: We have the subgroup S_5 generated by: $f = \left(\begin{array}{lllll} 1 & 2 & 3 & 4 & 5 \\ 2 & 1 & 3 & 4 & 5 \\ \end{array}\right) \qquad g = \left(\begin{array}{lllll} 1 & 2 & 3 & 4 & 5 \\ 1 & 2 & 4 & 5 & 3 \\ \end{array}\right)$ I must complete the other 3 elements for h,k, and l then do all the composition of every group to each other. I tried and obtained $h = \left(\begin{array}{lllll} 1 & 2 & 3 & 4 & 5 \\ 3 & 5 & 1 & 2 & 4 \\ \end{array}\right) \qquad k = \left(\begin{array}{lllll} 1 & 2 & 3 & 4 & 5 \\ 5 & 4 & 2 & 3 & 1 \\ \end{array}\right) \qquad l = \left(\begin{array}{lllll} 1 & 2 & 3 & 4 & 5 \\ 4 & 3 & 5 & 1 & 2 \\ \end{array}\right)$ but when I did $g \circ g$ and $f \circ g$ I obtained $g \circ g = \left(\begin{array}{lllll} 1 & 2 & 3 & 4 & 5 \\ 1 & 2 & 5 & 3 & 4 \\ \end{array}\right) \qquad f \circ g = \left(\begin{array}{lllll} 1 & 2 & 3 & 4 & 5 \\ 2 & 1 & 4 & 5 & 3 \\ \end{array}\right)$. I am very confused atm on what to do. Are $g \circ g$ and $f \circ g$ two of the elements that I need to find? So $g \circ g= h$ and $f \circ g= k$ and I will be able to figure out $l$? If that is the case, why do we have some of the same numbers being mapped two the same elements again? In g we have 1 -> 1 and in $g \circ g$ we have 1 -> 1. I don't think this is allowed. Any help will be graciously appreciated.

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