Page 110 - DMTH403_ABSTRACT_ALGEBRA

Unit 9: Direct Products

Definition: Let H and K be normal subgroups of a group G. We call G the internal direct product Notes
of H and K if
G = HK and H  K = {e}.
We write this fact as G = H × K.
For example, let us consider the familiar Klein 4-group

K = {e, a, b, ab}, where a = e, b = e and ab = ba.
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2
4
Let H = <a> and K = <b>. Then H  K = {e). Also, K = HK.
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 K = H × K.
4
Note that H  Z and K  Z 2  K  Z × Z .
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4
2
2
For another example, consider Z . It is the internal direct product of its subgroups H = {0, 5}
10
and K = {0, 2, 4, 6, 8}. This is because
(i) Z = H + K, since any element of Z is the sum of an element of H and an element of K, and
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10
(ii) H  K= {0} .
Now, can an external direct product also be an internal direct product? What does it say? It says
that the external product of G × G is the internal product (G × {e }) × ({e } × G ).
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2
1
2
2
1
We would like to make a remark here.
Remark: Let H and K be normal subgroups of a group G. Then the internal direct product of H
and K is isomorphic to the external direct product of H and K. Therefore, when we talk of an
internal direct product of subgroups we can drop the word internal, and just say ‘direct product
of subgroups’.
Let us now extend the definition of the internal direct product of two subgroups to that of
several subgroups.
Definition: A group G is the internal direct product of its normal subgroups H , H , .... ,H if
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n
2
(i) G = H H .... H and
n
1
2
(ii) H  H .... H , H .... H = {e}  i = 1, .... , n.
n
1
i
i+1
i-1
For example, look at the group G generated by {a, b, c}, where a = e = b = c and ab = ba, ac = ca,
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2
2
bc = cb. This is the internal direct product of < a >, < b > and < c >. That is G  Z × Z × Z .
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2
Now, can every group be written as an internal direct product of two or more of its proper
normal subgroups? Consider Z. Suppose Z = H × K, where H, K are subgroups of Z.
You know that H = < m > and K = < n > for some m, n  Z. Then mn  H  K. But if H × K is a direct
product, H  K = {0}. So, we reach a contradiction. Therefore, Z can’t be written as an internal
direct product of two subgroups.
By the same reasoning we can say that Z can’t be expressed as H × H × ..... × H , where Hi  Z
n
2
1
 i = 1, 2, .... , n.
When a group is an internal direct product of its subgroups, it satisfies the following theorem.
Theorem 1: Let a group G be the internal direct product of its subgroups H and K. Then
(a) each x  G can be uniquely expressed as x = hk, where h  H, k  K; and
(b) hk = kh  h  H , k  K .
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