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Section 16.3 The Isomorphism Theorems: Discovery guide

Subsection First Isomorphism Theorem

Most of the activities in this subsection will concern a homomorphism \(\funcdef{\varphi}{G}{G'}\) between some groups \(G\) and \(G'\text{.}\) As usual, write \(e\) for the identity element in \(G\) and \(e'\) for the identity element in \(G'\text{.}\) For convenience, write \(K\) and \(I\) to represent the kernel and image of \(\varphi\text{,}\) respectively.

Discovery 16.11. Kernel is normal.

Use Condition 2 of Fact 15.3.5 to verify that \(K\) is a normal subgroup of \(G\text{.}\)

Discovery 16.13.

Verify that \(\varphi\) is constant on each coset of \(K\) in \(G\text{.}\) That is, given two elements of \(G\) that lie in the same coset of \(K\text{,}\) the outputs of \(\varphi\) on those two elements will agree.

Discovery 16.13 lets us create an induced homomorphism

\begin{equation*} \funcdef{\induced{\varphi}}{G/K}{G'} \end{equation*}

by having \(\induced{\varphi}\) send each coset of \(K\) to the common output of \(\varphi\) on the elements in that coset. That is, we define \(\induced{\varphi}\) so that

\begin{equation*} \induced{\varphi}(x K) = \varphi(z) \end{equation*}

is true for every \(z\) in the coset \(x K\text{.}\) In particular, Discovery 16.13 lets us unambiguously write

\begin{equation*} \induced{\varphi}(x K) = \varphi(x) \text{,} \end{equation*}

because it tells us that it doesn't matter what element \(x\) we choose to represent the coset \(x K\text{,}\) as the output \(\varphi(x)\) doesn't vary on that coset.

Discovery 16.14.

Verify that \(\induced{\varphi}\text{,}\) as defined above, is a homomorphism. That is, verify the equality

\begin{equation*} \induced{\varphi}(x_1 K \cdot x_2 K) = \induced{\varphi}(x_1 K) \cdot \induced{\varphi}(x_2 K) \text{.} \end{equation*}

(On the left, your first task is to determine a representative element of the coset \(x_1 K \cdot x_2 K\) so that you may apply the definition of \(\induced{\varphi}\text{.}\))

Discovery 16.15.

(a)

Convince yourself that the image of \(\induced{\varphi}\) in \(G'\) is precisely the collection \(I\text{,}\) the image of \(\varphi\text{.}\)

(b)

Verify that \(\induced{\varphi}\) is a bijective correspondence between \(G/K\) and \(I\text{.}\) (In Task a, you convinced yourself that every element in \(I\) is an output for \(\induced{\varphi}\text{.}\) You must now check that each element in \(I\) cannot simultaneously be the output under \(\induced{\varphi}\) for two different input elements in \(G/K\text{.}\))

The activities to this point have now verified the following.

Discovery 16.16.

Using your list of kernels and images from Discovery 16.1, for each homomorphism in that activity, rewrite the conclusion of the First Isomorphism Theorem with the specifics of that example substituted in.

Subsection Second Isomorphism Theorem

Recall that the intersection of two collections is the collection of all elements that are common to both. In set theory we write \(A \cap B\) to denote the intersection of collections \(A\) and \(B\text{.}\) We have already seen that the intersection of two subgroups is again a subgroup (Discovery 5.9).

Discovery 16.18.

Rewrite the statement of the Second Isomorphism Theorem for the example of

\begin{align*} G \amp = \GL_2(\R), \amp H \amp = \Or_2(\R), \amp J \amp = \SL_2(\R) \text{,} \end{align*}

with the specifics of this example substituted in.

I will leave it up to you to consider the first two conclusions of the Second Isomorphism Theorem on your own or to read about them (afterwards) in the textbook. Let's instead focus on how the third conclusion is really an application of the First Isomorphism Theorem.

Discovery 16.19.

Assume \(H\text{,}\) \(J\text{,}\) and \(G\) are as in the assumptions of the Second Isomorphism Theorem. To apply the First Isomorphism Theorem, we need a homomorphism

\begin{equation*} \funcdef{\varphi}{H}{H J / J} \end{equation*}

with kernel \(H \cap J\text{.}\) So define

\begin{equation*} \varphi(h) = h J \text{.} \end{equation*}
(a)

Apply the result of Discovery 15.7 with \(A = J\) and \(B = H J\) to justify why we can consider the quotient group \(H J / J\text{.}\)

(b)

Elements of \(H J / J\) should be cosets of \(J\) in \(H J\text{,}\) which are of the form \(g J\) for \(g\) a product of an element of \(H\) with an element in \(J\) (in that order). But we have defined the output of \(\varphi(h)\) to be \(h J\text{,}\) where the coset representative is just an element of \(H\text{.}\) Explain why this is okay.

(c)

Verify that \(\varphi\) is a homomorphism.

(d)

Verify that the image of \(\varphi\) is all of \(H J / J\) as follows. Given a coset \(g J\) of \(J\) in \(H J\text{,}\) where \(g\) is a product of an element of \(H\) with an element in \(J\) (in that order), determine an element \(h\) in \(H\) so that

\begin{equation*} \varphi(h) = g J \text{.} \end{equation*}
(e)

Verify that the kernel of \(\varphi\) is \(H \cap J\) as follows. Given an element \(h\) of \(H\) so that \(\varphi(h)\) is the identity element in \(H J / J\) (refer to your answer to Discovery 15.11), demonstrate that \(h\) must also be an element of \(J\text{.}\)

Hint.

Fact 15.3.1, but applied to \(J\) not \(H\text{.}\)

Discovery 16.20.

What if \(H\) and \(J\) satisfy the conditions so that the internal product \(H J\) is isomorphic to the external product \(H \times J\text{?}\) (See Theorem 10.2 in the textbook, but ignore the \(H K = G\) part, because we are considering \(H K = H K\text{.}\)) What does the third conclusion of the Second Isomorphism Theorem say in this case?

Hint.

Discovery 15.10 will help you simplify one part of things.

Subsection Third Isomorphism Theorem

Discovery 16.21.

Take

\begin{align*} G \amp = D_8, \amp H \amp = \gen{r^2}, \amp J \amp = \gen{s, r^2} \text{.} \end{align*}

Clearly both \(H\) and \(J\) are subgroups of \(G\text{,}\) with \(H\) contained in \(J\text{.}\)

(a)

Verify that both \(H\) and \(J\) are normal in \(G\text{.}\)

(b)

List the elements of \(G / J\) by choosing representatives for each coset.

(c)

List the elements of \(G / H\) by choosing representatives for each coset. Then decide which of those cosets make up \(J / H\text{.}\)

(d)

Verify that the cosets making up \(J / H\) form a subgroup of \(G / H\text{.}\)

(e)

As \(J / H\) is a subgroup of \(G / H\text{,}\) the elements of \(G / H\) that are not in \(J / H\) can be partitioned into cosets of \(J / H\text{.}\) Perform this partitioning. (Careful: A representative a coset of \(J / H\) in \(G / H\) should be a coset of \(H\) in \(G\text{.}\) This is getting very meta!)

(f)

Use your partitioning of \(G / H\) into cosets of \(J / H\) to verify that \(J / H\) is normal in \(G /H\text{.}\)

Hint.

Discovery 15.13.

(g)

Finally, list the elements of \(\left. (G / H) \middle/ (J / H) \right. \) by choosing representatives for each coset. (Again, remember that these representatives should be cosets of \(H\) in \(G\text{.}\))

Compare this list with your list of elements of \(G / J\text{.}\) The Third Isomorphism Theorem says that these two groups should be isomorphic — does it appear that this is true?

Discovery 16.22.

Again, the Third Isomorphism Theorem is an application of the First Isomorphism Theorem. Let \(H\text{,}\) \(J\text{,}\) and \(G\) be as in the assumptions of the Third Isomorphism Theorem, and define

\begin{gather*} \funcdef{\varphi}{G / H}{G / J} \\ \varphi(x H) = x J \end{gather*}
(a)

For a definition like the one we've made for \(\varphi\text{,}\) it's important to verify that the result does not depend on the choice of coset representative. Perform this verification by checking that if \(x_1 H = x_2 H\) then also \(x_1 J = x_2 J\text{.}\)

Hint.

See Fact 15.3.1, and remember that \(H\) is assumed to be contained in \(J\text{.}\)

(b)

Verify that \(\varphi\) is a homomorphism.

(c)

Verify that the image of \(\varphi\) is all of \(G / J\text{.}\)

(d)

Verify that a coset of \(H\) in \(G\) is in the kernel of \(\varphi\) precisely when all of the elements it contains are elements from \(J\text{.}\) (This verifies that the kernel of \(\varphi\) is \(J / H\text{.}\))

Hint.

Remind yourself of Discovery 15.11 (applied to the quotient \(G / J\)) when you are deciding what it means for an element of \(G / H\) to be in the kernel of \(\varphi\text{.}\)