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226 ALGEBRA

5.8.1-2. Notion of a group.

A group is a set G with a composition law T satisfying the conditions:
1. The law T is associative.
2. There is a neutral element e G.
–1
3. For any a G, there is an inverse element a .
A group G is said to be commutative or abelian if its composition law T is commutative.
Example 2. The set Z of all integer numbers is an abelian group with respect to addition. The set of all
positive real numbers is an abelian group with respect to multiplication. Any linear space is an abelian group
with respect to the addition of its elements.
Example 3. Permutation groups.Let E be a set consisting of finitely many elements a, b, c, ... .A
permutation of E is a one-to-one mapping of E onto itself. A permutation f of the set E can be expressed in
the form
a b c ...
f = .
f(a) f(b) f(c) ...
On the set P of all permutations of E, the composition law is introduced as follows: if f 1 and f 2 are two
permutations of E, then their composition f 2 ◦ f 1 is the permutation obtained by consecutive application of f 1
and f 2. This composition law is associative. The set of all permutations of E with this composition law is a
group.
Example 4. The group Z 2 that consists of two elements 0 and 1 with the multiplication defined by
0 ⋅ 0 = 0, 0 ⋅ 1 = 1, 1 ⋅ 0 = 1, 1 ⋅ 1 = 1
and the neutral element 0 is called the group of modulo 2 residues.
Properties of groups:
–1
1. If aTa –1 = e,then a Ta = e.
2. eTa = a for any a.
3. If aTx = e and aTy = e,then x = y.
4. The neutral element e is unique.

5.8.1-3. Homomorphisms and isomorphisms.

Recall that a mapping f : A → B of a set A into a set B is a correspondence that associates
each element of A with an element of B.The range of the mapping f is the set of all
b B such that b = f(a). One says that f is a one-to-one mapping if it maps different
elements of A into different elements of B, i.e., for any a 1 , a 2 A such that a 1 ≠ a 2 ,we
have f(a 1 ) ≠ f(a 2 ).
A mapping f : A → B is called a mapping of the set A onto the set B if each element
of B is an image of some element of A, i.e., for any b B,there is a A such that b = f(a).
A mapping f of A onto B is said to be invertible if there is a mapping g : B → A such
that g(f(a)) = a for any a A. The mapping g is called the inverse of the mapping f and
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is denoted by g = f .
For definiteness, we use the multiplicative notation for composition laws in what follows,
unless indicated otherwise.

Let G be a group and let G be a set with a composition law. A mapping f : G → G is
2
2
called a homomorphism if
f(ab)= f(a)f(b) for all a, b G;

and the subset of G consisting of all elements of the form f(a), a G, is called a homo-
2
morphic image of the group G and is denoted by f(G). Note that here the set G with a
2
composition law is not necessarily a group. However, the following result holds.