Page 109 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes Can we define a binary operation on G by using the operations on G and G ? Let us try the
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method, namely, component-wise multiplication. That is, we define the operation * on G by
(a,b) * (c, d)=(a* c, b* d)  a, c  G , b, d  G .
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So, you have proved that G = G × G is a group with respect to *. We call G the external direct
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product of (G , * ) and (G , * ).
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For example, R is the external direct product of R with itself.
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Another example is the direct product (Z, +) × (R*, .) in which the operation is given by (m, X) *
(n, y) = (m + n, xy).
We can also define the external direct product of 3, 4 or more groups on the same lines.
Definition: Let (G , * ), (G , * ), . . . . . , (G,, * ) be n groups. Their external direct product is the
n
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group (G, *), where
G = G × G ..... × G and
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n
Thus, R is the external direct product of n copies of R,
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We would like to make a remark about notation now.
Remark: Henceforth, we will assume that all the operations *, * ,. . . , * are multiplication, unless
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mentioned otherwise. Thus, the operation on
G = G × G × ..... × G will be given by
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2
(a , ..... , a) . (b , ..... , b)
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= (a b , a b , .... , a b ,) V a , b  G . i
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i
a a
i
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Now, let G be the external direct product G × G . Consider the projection map
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T : G × G  G : ! : (x, y) = x.
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Then  is a group homomorphism, since
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 ((a, b) (c, d)) =  (ac, bd)
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= ac
=  (a, b)  (c, d)
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 is also onto, because any x  G is ! (x, e )
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Now, let us look at Ker  .
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Ker  = {(x, y)  G × G |  (x, y) = e }
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
= {(e , y) | y  Gz} = {e } × G .
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 {e} × G  G × G .
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Also, by the Fundamental Theorem of Homomorphism (G × G )/({e } × Gz)  G .
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We can similarly prove that G × {e }  G × G and (G × G )/(G × {e })  G .
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So, far we have seen the construction of G × G from two groups G and G . Now we will see
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under what conditions we can express a group as a direct product of its subgroups.
9.1.2 Internal Direct Product
Let us begin by recalling from Unit 5 that if H and K are normal subgroups of a group G, then HK
is a normal subgroup of G. We are interested in the case when HK is the whole of G. We have the
following definition.
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