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

Groups 15
notations (see the solutions) these tables coincide. For many purposes it is
natural to consider such groups of transformations as coinciding. Therefore
we shall consider abstract objects rather than sets of real elements (in
our case of transformations). Furthermore, we shall consider those binary
operations on arbitrary sets which possess the basic properties of the binary
operation in a group of transformations. Thus any binary operation
will be called a multiplication; if to the pair there corresponds the
element we call the product of and and we write In some
special cases the binary operation will be called differently, for example,
composition, addition, etc..
DEFINITION. A set G of elements of an arbitrary nature, on which
one can define a binary operation such that the following conditions are
satisfied, is called a group:
1) associativity : for any elements and of G;
2) in G there is an element such that for every element
of G; such element is called the unit (or neutral element) of group G;
3) for every element of G there is in G an element such that
such an element is called the inverse of element
From the results of Problems 12–14 we see that any group of transformations
is a group (in some sense the converse statement is also true
(see 55)). In this way we have already seen a lot of examples of groups.
All these groups contain a finite number of elements: such groups are
called finite groups. The number of elements of a finite group is called
the order of the group. Groups containing an infinite number of elements
are called infinite groups.
Let us give some examples of infinite groups.
EXAMPLE 7. Consider the set of all integer numbers. In this set
we shall take as binary operation the usual addition. We thus obtain
a group. Indeed, the role of the unit element is played by 0, because
for every integer Moreover, for every there exists
the inverse element (which is called in this case the opposite element),
because The associativity follows from the
rules of arithmetic. The obtained group is called the group of integers
under addition.
15. Consider the following sets: 1) all the real numbers; 2) all the
real numbers without zero. Say whether the sets 1 and 2 form a group
under multiplication.