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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.
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