Help reorder the alphabet 2

This problem can be solved using some basic facts about permutation groups .

The activity of rearranging letters of the alphabet is a permutation . The set of all permutations of 26 letters forms the permutation group \mathcal S {26}_. Two permutations $\sigma_1$ and $\sigma_2$ can be combined as $\sigma_2 \circ \sigma_1$ ( $\sigma_2$ after $\sigma_1$ ).

A cycle , written $(a_1\ a_2\ \dots\ a_i)$ is a permutation in which the element in position $a_1$ is moved to position $a_2$ , the element in position $a_2$ to position $a_3$ , and so on, and finally the element in position $a_i$ to position $a_1$ . Thus the cycle described in the problem is a 3-cycle.

A simple swapping of element is a 2-cycle. All permutations in $S_{26}$ (or any full permutation group) may be generated by combining 2-cycles. For instance, the 3-cycle $(1\ 2\ 3)$ may be generated as $(1\ 2)\circ(2\ 3)$ .

A permutation is called even if it can be generated by combining an even number of 2-cycles. They form a subgroup of $\mathcal S_{26}$ , called the alternating group ( $\mathcal A_{26}$ ). From the previous paragraph it is clear that a 3-cycle is an even permutation.

Any permutation that is not even is odd ; it can be generated by combining any arbitrary 2-cycle with some element of the alternating group.

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For this problem, note that the permutation needed to sort the alphabet may be odd. But since a 3-cycle is an even permutation, any combination of 3-cycles is even. Therefore, it may not be possible to sort the alphabet using 3-cycles.