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

44 Chapter 1
189. Inscribe in the dodecahedron five tetrahedra as explained above 10 .
Another proof of the non-solubility of the group consists in repeating
the argument of the proof of the non-solubility of the group of
rotations of the dodecahedron. To do this one must solve the next problem.
190. Prove that every even permutation of degree 5, different from
the identity, can be decomposed into independent cycles in just one of
the following ways: a) b) c)
191. Let N be a normal subgroup of group Prove that if N
contains at least one permutation which splits into independent cycles
in one of the ways indicated in Problem 190, then N contains all the
permutations splitting into independent cycles in this way.
192. Prove that the group does not contain normal subgroups
except the identity and the whole group.
From the results of Problems 192, 161 and from the group being
not commutative, it follows that the group is not soluble.
193. Prove that the symmetric group for contains a subgroup
isomorphic to
From the results of Problems 193 and 162 we obtain the following
theorem.
THEOREM 5. For the symmetric group is not soluble.
The proof of this theorem, as well as the other results of this chapter,
will be needed in the next chapter to demonstrate the non-solvability by
radicals of algebraic equations of degree higher than four 11 .
10 To inscribe the 5 tetrahedra inside the dodecahedron one can start from the 5
Kepler cubes. For their description and their relation with the tetrahedra see the
footnote of the solution of Problem 189. (Translator’s note)
11 The following books are indicated to students who desire to study the theory
of groups more deeply: Kargapolov M.I., Merzlyakov Y.I., (1972), Fundamentals of
the Theory of Groups, Graduate Texts in Mathematics, (Springer-Verlag: New York);
Vinberg. E.B., (2003), A Course in Algebra, Graduate Studies in Mathematics, v. 56,
(AMS).