# Direct Producrs and Quotient Groups

• Nov 21st 2013, 01:09 AM
Bernhard
Direct Producrs and Quotient Groups
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 $\displaystyle ( \mathbb{Z}_6 \times \mathbb{Z}_4 ) / (2,2)$

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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 $\displaystyle ( \mathbb{Z}_6 \times \mathbb{Z}_4 ) / (2,2)$ but had some difficulties, when I realised that I was unsure of whether the group $\displaystyle ( \mathbb{Z}_6 \times \mathbb{Z}_4 )$ was a group under multiplication or addition. SO essentially I did not know how to carry out group operations in $\displaystyle ( \mathbb{Z}_6 \times \mathbb{Z}_4 ) / (2,2)$.

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 $\displaystyle G_1$ and $\displaystyle G_2$ be groups. The set of all ordered pairs $\displaystyle (x_1, x_2)$ such that $\displaystyle x_1 \in G_1$ and $\displaystyle x_2 \in G_2$ is called the direct product of $\displaystyle G_1$ and $\displaystyle G_2$, denoted by $\displaystyle G_1 \times G_2$.

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Then Proposition 3,3,4 reads as follows:

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3,3,4 Proposition. Let $\displaystyle G_1$ and $\displaystyle G_2$ be groups.
(a) The direct product $\displaystyle G_1 \times G_2$ is a group under the operation defined for all $\displaystyle (a_1, a_2) , (b_1, b_2) \in G_1 \times G_2$ by

$\displaystyle (a_1, a_2) (b_1, b_2) = (a_1b_1, a_2b_2 )$.

(b) etc etc

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However in Example 3.3.3 on page 119 we find the group $\displaystyle ( \mathbb{Z}_2 \times \mathbb{Z}_2 )$ dealt with as having addition as its operation.

My question is - what is the convention on direct products of $\displaystyle ( \mathbb{Z}_n \times \mathbb{Z}_m )$ - 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
• Nov 24th 2013, 02:31 AM
Haven
Re: Direct Producrs and Quotient Groups
In general $\displaystyle \mathbb{Z}_n$ is not a group under multiplication. For example $\displaystyle \mathbb{Z}_6$ is not a group under multiplication, as 2*3=0. However the group of units mod 6 $\displaystyle U(6) = \{1,5\}$ IS a group under multiplication.

I feel that the book is using the same notation as Gallian's Contemporary Abstract Algebra. Where $\displaystyle \mathbb{Z}_n$ denotes the group under addition.

So in this case, it's useful to compute the order of (2,2). The order of 2 in $\displaystyle \mathbb{Z}_6$ is 3 and the order of 2 in $\displaystyle \mathbb{Z}_4$ is 2. So the order of (2,2) is the least common multiple of 2 and 3, which is 6. As $\displaystyle \mathbb{Z}_6\times\mathbb{Z}_4$ is a group of order 24, the factor group $\displaystyle \mathbb{Z}_6\times\mathbb{Z}_4/\langle(2,2)\rangle$will be of order $24/6 = 4$. So just find a representative for each coset and you'll "compute the factor group"