Why Read This Book? - Index of

Thus
f 2 = f ◦ f = (132)(56) ◦ (132)(56) = (123)
f 3 = f 2 ◦ f = (123) ◦ (132)(56) = (56)
f 4 = f 3 ◦ f = (56) ◦ (132)(56) = (132)
f 5 = f 4 ◦ f = (132) ◦ (132)(56) = (123)(56)
f 6 = f 5 ◦ f = (123)(56) ◦ (132)(56) = (1)
8.4 Permutation Groups 271
(f ) ={(1), f, f 2 ,f 3 ,f 4 ,f 5 }
={(1), (132)(56), (123), (56), (132), (123)(56)}
EXERCISE 8.4.5 In S6, determine the subgroup generated by (125)(346).
(8.55)
(8.56)
EXERCISE 8.4.6 For g = (12) and h = (34) in S4, determine ({g, h}), the
subgroup of S4 generated by g and h.
EXERCISE 8.4.7 In S4, let f = (123) and g = (14). Determine whether g
generates the same left and right cosets of the subgroup generated by f .
8.4.3 The Alternating Group
Now let’s look at an important subgroup of Sn. A permutation of the form (ij) is
called a transposition because all it does is switch the positions of two elements.
Notice (ij) −1 = (ij) and (ij) = (ji). Ifi = 1 (or j = 1, it doesn’t matter), then (ij)
becomes (1j). If neither i nor j is 1, then
(ij) = (1i)(1j)(1i) (8.57)
By Eq. (8.57), every transposition that does not involve 1 can be written as a
product of three transpositions, each of which involves 1. So if you’re playing
some game where objects are lined up in positions 1,...,nand you want to swap
the positions of the objects in positions i and j, it is possible to do it even if you
restrict yourself to swapping an object’s position only with the object in position 1.
It just takes three moves instead of one. But notice that both (ij) and its equivalent
in (8.57) use an odd number of transpositions.
Even though (ijk) is not a transposition, it can be written as a composition of
two transpositions. One way to do it is
(ijk) = (ik)(ij) (8.58)
If none of {i, j, k} is 1, what does (8.58) become if we require that all transpositions
involve 1, as in (8.57)? Taking a hint from Eq. (8.57) and applying it to both (ik)
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