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228 ALGEBRA
THEOREM 1. If f is a homomorphism of a group G onto a group G and H is the set
of all elements of G that are mapped to f(e) (e is the identity element of G), then H is a
normal subgroup in G.
THEOREM 2(ON GROUP HOMOMORPHISMS). If f is a homomorphism of a group G onto
a group G and H is the normal subgroup of G consisting of the elements mapped to the
identity element of G, then the group G and the quotient group G/H are isomorphic.
Thus, given a homomorphism f of a group G onto a group G,the kernel H of the
2
homomorphism is a normal subgroup of G, and conversely any normal subgroup H of G is
the kernel of the homomorphism of G onto the quotient group G/H.
Remark. Given a homomorphism of a group G onto a set G, all elements of the group G are divided into
mutually disjoint classes, each class containing all elements of G that are mapped into the same element of G.
n
Example 6. Let R be the n-dimensional linear coordinate space, which is an abelian group with respect
to addition of its elements. This space is the direct product of one-dimensional spaces:
1
1
n
R = R (1) ⊗ ·· · ⊗ R (n) .
1
1
n
Since R (n) is an abelian subgroup, the set R (n) is a normal subgroup of the group R . The coset corresponding
1
n
to an element a R is the straight line passing through a in the direction parallel to the straight line R (n) ,and
1
n
the quotient group R /R (n) is isomorphic to the (n – 1)-dimensional space R n–1 :
n–1 n 1 1 1
R = R /R (n) = R (1) ⊗ ··· ⊗ R (n–1) .
5.8.2. Transformation Groups
5.8.2-1. Group of linear transformations. Its subgroups.
Let V beareal finite-dimensional linear space and let A : V → V be a nondegenerate linear
operator. This operator can be regarded as a nondegenerate linear transformation of the
space V ,since A maps different elements of V into different elements, and for any y V
there is a unique x V such that Ax = y.
The set of all nondegenerate linear transformations A of the n-dimensional real linear
space V is denoted by GL(n).
The product AB of linear transformations A and B in GL(n)isdefined by the relation
(AB)x = A(Bx) for all x V .
This product is a composition law on GL(n).
THEOREM. The set GL(n) of nondegenerate linear transformations of an n-dimensional
real linear space V with the above product is a group.
The group GL(n) is called the general linear group of dimension n.
A subset of GL(n) consisting of all linear transformations A such that det A = 1 is a
subgroup of GL(n) called the special linear group of dimension n and denoted by SL(n).
A sequence {A k } of elements of GL(n)is said tobe convergent to an element A GL(n)
as k →∞ if the sequence {A k x} converges to Ax for any x V .
Types of subgroups of GL(n):
1. Finite subgroups are subgroups with finitely many elements.
2. Discrete subgroups are subgroups with countably many elements.
3. Continuous subgroups are subgroups with uncountably many elements.