Section 15.2 Quotients: Discovery guide
Our goal is to use the group operation on \(G\) to try to make a quotient \(G/H\) (where \(H\) is a subgroup of \(G\)) into a group as well.
A group needs a group operation. We have an operation to multiply (or add, as the case may be) together elements of \(G\text{.}\) How will we multiply elements of \(G/H\) together, when those quotient elements are cosets? How will we multiply a whole collection of elements against another whole collection of elements?
Well, we already know how to multiply two subgroups together, from our investigation of internal group products. Let's use the same sort of idea: given subgroup \(H\) and elements \(x\) and \(y\) of a group \(G\text{,}\) define the product of (left) cosets
to be the collection of all results of computing a product of an element from \(x H\) with an element from \(y H\) (in that order).
Discovery 15.1.
Consider \(G = \Z\) and \(H = \gen{4}\text{.}\) The cosets of \(H\) are listed in Example 15.1.4.
(a)
Compute each of the following.
Notes:
Here we are using additive notation.
Your answers for each computation should be a list of integers, just as each coset is written as a list of integers in Example 15.1.4.
Make sure you are considering all possible ways of adding an integer from the first coset with an integer from the second coset, but don't include duplicate results from these computations.
(i)
\(H + H \text{.}\)
(ii)
\(H + (1 + H) \text{.}\)
(iii)
\(H + (2 + H) \text{.}\)
(iv)
\(H + (3 + H) \text{.}\)
(v)
\((1 + H) + (1 + H) \text{.}\)
(vi)
\((1 + H) + (2 + H) \text{.}\)
(vii)
\((1 + H) + (3 + H) \text{.}\)
(viii)
\((2 + H) + (2 + H) \text{.}\)
(ix)
\((2 + H) + (3 + H) \text{.}\)
(x)
\((3 + H) + (3 + H) \text{.}\)
(Since \(\Z\) is Abelian, there is no need to consider computations like \((3 + H) + (1 + H) \text{.}\))
(b)
Do you notice any patterns in your computations?
Discovery 15.2.
Recall that
with
is the group of rotational symmetries of a thin hexagonal plate.
Consider the subgroup \(H = \gen{r^3}\text{.}\)
(a)
Write out the elements of each left coset of \(H\text{.}\) Make sure you are simplifying your coset element expressions to one of the twelve element forms listed for \(D_6\) in (✶).
(b)
How many elements are in the quotient \(G/H\text{?}\)
(c)
Compute each of the following.
Notes:
Your answers for each computation should be a list of elements of \(D_6\text{.}\)
Make sure you are considering all possible ways of multiplying a \(D_6\) element from the first coset against a \(D_6\) element from the second coset, but don't include duplicate results from these computations.
(i)
\(s H \cdot s H \text{.}\)
(ii)
\(r H \cdot r H \text{.}\)
(iii)
\(r H \) times your result from Task c.ii (in either order, doesn't matter).
(iv)
\(s H \cdot r H \text{.}\)
(d)
Do the patterns of the computations in Task c look familiar? Compare these computation patterns with the results of Discovery 10.13 and Discovery 10.14.
Discovery 15.3.
Working with \(D_6\) again (see the introduction to Discovery 15.2 above), but this time using subgroup \(H = \gen{s}\text{.}\)
(a)
Write out the elements of each left coset of \(H\text{.}\) Make sure you are simplifying your coset element expressions to one of the twelve element forms listed for \(D_6\) in (✶).
(b)
How many elements are in the quotient \(G/H\text{?}\)
(c)
Compute \(s H \cdot r H\) using the same instructions as in the previous activities. Did the result work out the way you expected, given your experience with other examples so far in this discovery set?
In your follow-up textbook reading assignment in Chapter 15, you will read about why some of the examples worked, but the last one didn't.