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

Groups 43
180. Prove that by multiplying an even permutation by an arbitrary
transposition one obtains an odd permutation, and, conversely, by multiplying
an odd permutation with an arbitrary transposition one obtains
an even permutation.
181. Prove that an even permutation can be decomposed only into
a product of an even number of transpositions, and an odd permutation
only into an odd number of transpositions.
182. Determine the parity of an arbitrary cycle of length: a) 3; b) 4;
c) m.
183. Prove that the result of the multiplication of two permutations
of the same parity is an even permutation, whereas the result of the multiplication
of two permutations of opposite parities is an odd permutation.
184. Let be an arbitrary permutation. Prove that and have
the same parity.
From the results of Problems 183 and 184 it follows that the set of
all the even permutations form a subgroup of group
DEFINITION. The group of all even permutations of degree is called
the alternating group of degree and it is denoted by
185. Prove that for is not commutative.
186. Prove that the alternating group is a normal subgroup of the
symmetric group and find the partition of by
187. Calculate the number of elements of the group
188. Prove that the groups and are soluble.
We now prove that the alternating group is not soluble. One of
the possible proofs uses the following construction. We inscribe in the
dodecahedron five regular tetrahedra, numbered by the numbers 1, 2, 3,
4 and 5 in such a way that to every rotation of the dodecahedron there
corresponds an even permutation of the tetrahedra, and that to different
rotations there correspond different permutations. So we have defined an
isomorphism between the group of rotations of the dodecahedron and the
group of the even permutations of degree 5. The non-solubility of the
group will thus follow from the non-solubility of the group of rotations
of the dodecahedron.