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

Groups 19
35. Suppose that is the maximal common divisor of the integers
and and that has order Prove that the element has order
36. Find all generators of the group of rotations of the regular dodecagon.
37. Let be an element of infinite order. Prove that the elements
are all distinct.
DEFINITION. If is an element of infinite order and group G has no
other elements but then G is called an infinite
cyclic group and its generator.
38. Prove that the group of the integers is a cyclic group under
addition (see Example 7, §1.3). Find all generators.
EXAMPLE 9. Let be an integer different from zero. Consider all
the possible remainders of the division of integers by i.e., the numbers
Let us introduce in this set of remainders the following
binary operation. After adding two remainders as usually, we keep the
remainder of the division by of the obtained sum. This operation is
called the addition modulo So we have, summing modulo 4, 1 + 2 = 3,
but 3 + 3 = 2.
39. Write the multiplication table for the addition modulo: a) 2; b)
3; c) 4.
40. Prove that the set of remainders with the addition modulo form
a group, and that this group is a cyclic group of order
Consider again an arbitrary cyclic group of order
41. Prove that where and
if and only if modulo one has
From the result of the preceding problem it follows that to the multiplication
of the elements in an arbitrary cyclic group there corresponds
the addition of the remainders modulo Similarly to the multiplication
of two elements in an infinite cyclic group there corresponds the addition
of integers (see 7). We come in this way to an important notion in the
theory of groups: the notion of isomorphism.
1.5 Isomorphisms
DEFINITION. Let two groups and be given with a bijective mapping