Abstract Algebra is a branch of mathematics that deals with the study of algebraic structures, such as groups, rings, and fields, and their properties and relationships. The notes of Abstract Algebra typically cover topics such as groups and their subgroups, cyclic groups, normal subgroups, factor ...
Learning outcomes: 1. External direct product of groups.
2. Product of cyclic groups.
3. Order of an element in the product of two groups.
4. Quotient of direct products.
In this module we de ne a suitable binary operation on the cartesian product G1 G2 of two
groups G1 and G2 so that (G1 G2; ) is a group which we call the external direct product of
groups. This is a widely used technique to construct a new group from old as well as to
decompose a group as a product of relatively better known groups. Thus external direct
product of groups is an important notion in the structure theory of nite groups.
Let G1; G2; ; Gn be n groups. De ne a binary operation on the cartesian product G =
G1 G2 Gn by:
(a1; a2; ; an) (b1; b2; ; bn) = (a1b1; a2b2; ; anbn) (0.1)
for every (a1; a2; ; an); (b1; b2; ; bn) 2 G. Here aibi is the product of two elements ai and bi in Gi
for every i = 1; 2; ; n.
For example, the binary operation on K4 Z is given by:
(a; m) (b; n) = (ab; m + n); for every (a; m); (b; n) 2 K4 Z:
It is easy to check that (K4 Z; ) is a group having the identity element (e; 0) and for every (a; n)
2 K4 Z, (a; n)1 = (a; n).
Similarly (G; ) is a group with the identity element e = (e1; e2; ; en) and (a1; a2; ; an)1 =
(a11 ; a12 ; ; a1n ) for every (a1; a2; ; an) 2 G where ei is the identity element of Gi for every
i = 1; 2; ; n.
De nition 0.1. Let G1; G2; ; Gn be n groups and G = G1 G2 Gn. Then the group (G; ) is called
the external direct product of the groups G1; G2; ; Gn.
Example 0.2. Consider the noncommutative group (S3; ) and the in nite group (Z; +). Then the
binary operation on G = S3 Z is given by:
( ; m) ( ; n) = ( ; m + n):
2
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