# Exact Sequences - Diagrams that 'commute' - Example

• Aug 17th 2013, 11:36 PM
Bernhard
Exact Sequences - Diagrams that 'commute' - Example
I am reading Dummit and Foote Section 10.5 on Exact Sequences.

I am trying to understand Example 1 as given at the bottom of page 381 and continued at the top of page 382 - please see attachment for the diagram and explanantion of the example.

The example, as you can no doubt see, requires an understanding of the nature of the quotient module \$\displaystyle (\mathbb{Z} / m \mathbb{Z} ) / (n \mathbb{Z} / m \mathbb{Z} ) \$

To make this quotient more tangible, in this example take m = 6, n = 3 so k = 2.

Then we are trying to understand the nature of the quotient module \$\displaystyle (\mathbb{Z} / 6 \mathbb{Z} ) / (3 \mathbb{Z} / 6 \mathbb{Z} ) \$

Now consider the nature of \$\displaystyle (\mathbb{Z} / 6 \mathbb{Z} ) \$

We have \$\displaystyle 0 + \mathbb{Z} / 6 \mathbb{Z} \$ = { ... ... -18, -12, -6, 0 , 6, 12, 18, 24, ... ... }

and \$\displaystyle 1 + \mathbb{Z} / 6 \mathbb{Z} \$ = {... ... -17, -11, -5, 1, 7, 13, 19, 25, ... }

and so on

But what is \$\displaystyle 3 \mathbb{Z} / 6 \mathbb{Z} \$ ? and indeed, further, what is \$\displaystyle (\mathbb{Z} / 6 \mathbb{Z} ) / (3 \mathbb{Z} / 6 \mathbb{Z} ) \$ ?

Peter
• Aug 18th 2013, 01:34 AM
agentmulder
Re: Exact Sequences - Diagrams that 'commute' - Example
I am an amateur but i find your question interesting. I thought Z/6Z = {0 , 1 , 2 , 3 , 4 , 5} ?

That would make 3Z/6Z = {0 , 3} ?

So (Z/6Z)/(3Z/6Z) = {0, 1, 2, 3, 4, 5}/{0, 3} = {1 , 2 , 4 , 5} <---- really not sure about this but it appears the parenthesis force the division in the center to become a SUBTRACTION of sorts ... {0, 1, 2 , 3, 4 , 5} - {0 , 3} = {1 , 2 , 4 , 5} where the elements 0 and 3 have been subtracted from Z/6Z.

I apologize if this is way off.