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

20 Chapter 1
from into (see §1.2) with the following property: if
and in group then in group
In other words, to the multiplication in there corresponds under
the multiplication in The mapping is thus called an isomorphism
between groups and and the groups and are said to be
isomorphic. The condition for a bijective mapping to be an isomorphism
can also be expressed by the following condition: for
all elements and of group here the product is taken in the group
and the product in the group
42. Which amongst the following groups are isomorphic: 1) the group
of rotations of the square; 2) the group of symmetries of the rhombus;
3) the group of symmetries of the rectangle; 4) the group of remainders
under addition modulo 4?
43. Let be an isomorphism. Prove that the inverse
mapping is an isomorphism.
44. Let and be two isomorphisms. Prove
that the compound mapping is an isomorphism.
From the two last problems it follows that two groups which are isomorphic
to a third group are isomorphic to each other.
45. Prove that every cyclic group of order is isomorphic to the
group of the remainders of the division by under addition modulo
46. Prove that every infinite cyclic group is isomorphic to the group
of integers under addition.
47. Let be an isomorphism. Prove that where
and are the unit elements in groups G and F.
48. Let be an isomorphism. Prove that
for every element of group G.
49. Let be an isomorphism and let Prove that
and have the same order.
If we are interested in the group operation and not in the nature of
the elements of the groups (which, in fact, does not play any role), then
we can identify all groups which are isomorphic. So, for example, we shall
say that there exists, up to isomorphism, only one cyclic group of order
(see 45), which we denote by and only one infinite cyclic group (see
46), which we indicate by