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In Beachy and Blair: Abstract Algebra, Section 3.8 Cosets, Normal Groups and Factor Groups, Exercise 17 reads as follows:
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17. Compute the factor group ( \mathbb{Z}_6 \times \mathbb{Z}_4 ) / (2,2)
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Since I did not know the meaning of "Compute the factor group" I proceeded to try to list them members of ( \mathbb{Z}_6 \times \mathbb{Z}_4 ) / (2,2) but had some difficulties, when I realized that I was unsure of whether the group ( \mathbb{Z}_6 \times \mathbb{Z}_4 ) was a group under multiplication or addition. SO essentially I did not know how to carry out group operations in ( \mathbb{Z}_6 \times \mathbb{Z}_4 ) / (2,2).
Reading Beachy and Blair, Chapter 3 Groups, page 118 (see attachment) we find the following definition:
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3,3,3 Definition. Let G_1 and G_2 be groups. The set of all ordered pairs (x_1, x_2) such that x_1 \in G_1 and x_2 \in G_2 is called the direct product of G_1 and G_2, denoted by G_1 \times G_2.
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Then Proposition 3,3,4 reads as follows:
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3,3,4 Proposition. Let G_1 and G_2 be groups.
(a) The direct product G_1 \times G_2 is a group under the operation defined for all (a_1, a_2) , (b_1, b_2) \in G_1 \times G_2 by
(a_1, a_2) (b_1, b_2) = (a_1b_1, a_2b_2 ).
(b) etc etc
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However in Example 3.3.3 on page 119 we find the group ( \mathbb{Z}_2 \times \mathbb{Z}_2 ) dealt with as having addition as its operation.
My question is - what is the convention on direct products of ( \mathbb{Z}_n \times \mathbb{Z}_m ) - does one use addition or multiplication?
Presumably, since the operations involve integers the matter is more than one of notation?
Can someone please clarify this matter?
Would appreciate some help.
Peter
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17. Compute the factor group ( \mathbb{Z}_6 \times \mathbb{Z}_4 ) / (2,2)
----------------------------------------------------------------------------------------------------------------------
Since I did not know the meaning of "Compute the factor group" I proceeded to try to list them members of ( \mathbb{Z}_6 \times \mathbb{Z}_4 ) / (2,2) but had some difficulties, when I realized that I was unsure of whether the group ( \mathbb{Z}_6 \times \mathbb{Z}_4 ) was a group under multiplication or addition. SO essentially I did not know how to carry out group operations in ( \mathbb{Z}_6 \times \mathbb{Z}_4 ) / (2,2).
Reading Beachy and Blair, Chapter 3 Groups, page 118 (see attachment) we find the following definition:
-------------------------------------------------------------------------------------------------------------
3,3,3 Definition. Let G_1 and G_2 be groups. The set of all ordered pairs (x_1, x_2) such that x_1 \in G_1 and x_2 \in G_2 is called the direct product of G_1 and G_2, denoted by G_1 \times G_2.
----------------------------------------------------------------------------------------------------------------
Then Proposition 3,3,4 reads as follows:
-----------------------------------------------------------------------------------------------------------------
3,3,4 Proposition. Let G_1 and G_2 be groups.
(a) The direct product G_1 \times G_2 is a group under the operation defined for all (a_1, a_2) , (b_1, b_2) \in G_1 \times G_2 by
(a_1, a_2) (b_1, b_2) = (a_1b_1, a_2b_2 ).
(b) etc etc
------------------------------------------------------------------------------------------------------------------
However in Example 3.3.3 on page 119 we find the group ( \mathbb{Z}_2 \times \mathbb{Z}_2 ) dealt with as having addition as its operation.
My question is - what is the convention on direct products of ( \mathbb{Z}_n \times \mathbb{Z}_m ) - does one use addition or multiplication?
Presumably, since the operations involve integers the matter is more than one of notation?
Can someone please clarify this matter?
Would appreciate some help.
Peter