Abel's theorem in problems and solutions - School of Mathematics

Solutions 143
180. Since
the lower row of the product is obtained from the lower row of the initial
permutation interchanging numbers and By virtue of the result of
Problem 178 the permutation obtained and the initial permutation have
different parities.
181. Every permutation splits into a product of transpositions (see
174, 175). Suppose a permutation be decomposed into a product of
transpositions: We can write
being the identity permutation. Since the permutation is even and is
multiplied times by a transposition, one obtains (see 180) that if is
even is an even permutation, and if is odd is an odd permutation.
182. See the hint to Solution 175. Answer. a) even; b) odd; c) even
for odd, odd for even.
183. Hint. Decompose the given permutations into products of transpositions
(see 181). Count the number of transpositions in these products.
184. If the permutations should have different parities, then (see 183)
the permutation should be odd, which is not true.
185. For example,
186. If is an even permutation then is also even, independently
of the parity of the permutation Hence is a normal subgroup
of the group Let be an odd permutation. We prove that the coset
contains all the odd permutations. Let be an odd permutation.
Thus is even. It follows that the permutation belongs to
the coset Therefore the group is decomposed, by the subgroup
into two cosets: that of all the even permutations and that of all the
odd permutations.