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

Groups 13
7. Find all symmetries of the rectangle and write their multiplication
table.
1.2 Groups of transformations
Let X and Y be two sets of elements of arbitrary nature, and suppose that
every element of X is put into correspondence with a defined element
of Y. Thus one says that there exists a mapping of the set X into
the set The element is called the image of the element
and the pre-image of element One writes:
DEFINITION. The mapping is called surjective (or, equivalently,
a mapping of set X onto set Y) if for every element of Y there
exists an element of X such that i.e., every of Y has a
pre-image in X.
8. Let the mapping put every capital city in the world in correspondence
with the first letter of its name in English (for example,
= L). Is a mapping of the set of capitals onto the entire English alphabet?
DEFINITION. The mapping is called a one to one (or
bijective) mapping of the set X into the set Y if for every in Y there
exists a pre-image in X and this pre-image is unique.
9. Consider the following mappings of the set of all integer numbers
into the set of the non-negative integer numbers:
Which amongst these mappings are surjective, which are bijective?
Let M be an arbitrary set. For brevity we shall call any bijective
mapping of M into itself a transformation of set M.
Two transformations and will be considered equal if
for every element A of M. Instead of term ‘transformation’ the
term permutation is often used. We shall use this term only when the
transformation is defined on a finite set. A permutation can thus be
written in the form