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} ) $ ?

Can someone please help clarify this matter?

Peter