Why Read This Book? - Index of

262 Chapter 8 Groups
EXERCISE 8.3.1 Finish the proof that Z/(n) with operation ⊕n is a group with
the following steps.
(a) Show ⊕n is well defined.
(b) Show ⊕n is associative.
(c) Determine the identity element.
(d) For an arbitrarily chosen [k], determine the inverse of [k].
And we are done. Table 8.36 displays the final product for the case Z/(6).
⊕n [0] [1] [2] [3] [4] [5]
[0] [0] [1] [2] [3] [4] [5]
[1] [1] [2] [3] [4] [5] [0]
[2] [2] [3] [4] [5] [0] [1]
[3] [3] [4] [5] [0] [1] [2]
[4] [4] [5] [0] [1] [2] [3]
[5] [5] [0] [1] [2] [3] [4]
(8.36)
Let’s take a closer look at the equivalence classes [0], [1],...,[n − 1], for
there is another standard notation for these sets that we will use below when we
derive quotient groups in general. Though we know from Eq. (8.33) what is in
each equivalence class, let’s consider a way to visualize an arbitrary equivalence
class [k] in terms of the subgroup of the integers (n) =[0] that motivated this
whole mess in the first place.
Picture all the integers on the number line in standard fashion, but pretend
that each integer is like a key on a piano keyboard. Take countably infinitely
many of your friends, and each of you place a finger on the elements of (n) =[0],
so that you are pointing to all the multiples of n: {...,−2n, −n, 0,n,2n,...}. Now
suppose you want to point to all the elements of some equivalence class [k]. How
can you do it? Everyone in unison should lift his or her finger off the keyboard,
and everyone should shuffle over to the right k units, then put his or her finger back
down. In other words, to generate all the elements of [k], take all the elements of
(n) and add k to each one, so that you are translating the entire set (n) through
the integers by k units. Notice in this imagery, there will always be some person
pointing to one of the integers {0, 1, 2,...,n− 1}. Here is another way to describe
what we have done.
[k] ={x + k : x ∈ (n)} (8.37)
Let’s create new notation for the construction in Eq. (8.37), writing
(n) + k ={x + k : x ∈ (n)} (8.38)
This is a slight abuse of orthodox notation because it appears that we are combining
the subgroup (n) with an integer k using integer addition. This notation