UA-AUG-AUMAT229 Normal subgroups and quotient groups: Discovery guide

Using the pattern of conjugacy classes in $D_{6}$ from Discovery 6 of Discovery 14.4, check which of the following subgroups of $D_{6}$ satisfy Definition 15.3.2.

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(a)

$⟨ r^{3} ⟩ .$

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(b)

$⟨ r ⟩ .$

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(c)

$⟨ s ⟩ .$

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Discovery 15.6.

Repeat Discovery 15.5, but this time use Condition 5 of Fact 15.3.5. (You may wish to distribute the coset-computing duties amongst your group-mates.)

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Discovery 15.7.

Suppose $A$ and $B$ are subgroups in a group $G,$ with $A$ entirely contained in $B .$ Prove that if $A$ is normal in $G,$ then $A$ is also normal in $B .$

Careful: It's not as simple as just applying the definition of normal, because conjugacy classes in $B$ will be different from conjugacy classes in $G .$

Hint.

Apply Condition 2 of Fact 15.3.5 twice, first to the assumption that $A$ is normal in $G$ (using conjugates $g^{- 1} a g$ for $a$ in $A$ and $g$ in $G$ ), and second to the desired conclusion that $A$ is normal in $B$ (using conjugates $b^{- 1} a b$ for $a$ in $A$ and $b$ in $B$ ).

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Discovery 15.8.

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(a)

Explain why Condition 1 of Fact 15.3.5 is equivalent to verifying Definition 15.3.2.

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(b)

Explain why Condition 2 of Fact 15.3.5 is equivalent to verifying Condition 1.

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Discovery 15.9.

In this activity we will explore what happens if we “divide out” by the whole group.

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(a)

Verify that $G$ is always a normal subgroup of itself.

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(b)

How many cosets of $G$ are there? So how many elements does the group $G / G$ have?

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(c)

What “form” does the group $G / G$ take? That is, to what group is it isomorphic?

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Discovery 15.10.

In this activity we will explore what happens if we “divide out” by the trivial subgroup.

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(a)

Verify that ${e}$ is always a normal subgroup of the group in which it resides.

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(b)

How many cosets of ${e}$ are there? So how many elements does the group $G / {e}$ have?

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(c)

What “form” does the group $G / {e}$ take? That is, to what group is it isomorphic?

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Discovery 15.11.

When $H$ is a normal subgroup of $G,$ which coset of $H$ acts as the identity element in the quotient group $G / H ?$

Hint.

Re-read the identity group axiom to remind yourself of the defining properties of the identity element in a group, and then consider equality (✶) in Fact 15.3.3.

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Discovery 15.12.

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(a)

Verify that every subgroup of an Abelian group is normal.

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(b)

Using equality (✶) in Fact 15.3.3, verify that if $G$ is Abelian and $H$ is a subgroup (normal by Task a), then

x H \cdot y H and y H \cdot x H

always evaluate to the same result. (This verifies that $G / H$ is always Abelian when $G$ is Abelian.)

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Discovery 15.13.

Recall that the index of a subgroup is the number of cosets. Suppose $H$ is a subgroup of index two. Without peeking in the textbook, use Condition 5 of Fact 15.3.5 to prove that $H$ must be normal.

Hint.

The left cosets of $H$ partition $G .$ And so do the right cosets. Compare these two partitions.

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Discovery 15.14.

Use the result of Discovery 15.13 to verify that each of the following subgroups is normal in the stated parent group.

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(a)

$A_{n}$ in $S_{n} .$

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(b)

$⟨ r ⟩$ in $D_{n} .$

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(c)

${SO}_{n} (R)$ in $O_{n} (R) .$

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Discovery 15.15.

In this activity we will practise analyzing a quotient group in a specific example. Consider $G = D_{n},$ wherein we have algebra rules

\begin{aligned} r^{n} & = e, & s^{2} & = e, & s r^{k} = r^{n - k} s . \end{aligned}

Assume that $n$ is even, and let $H = ⟨ r^{2} ⟩ .$

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(a)

We know $| G | = 2 n$ and $| r^{2} | = n / 2 .$ So how many left cosets of $H$ in $G$ must there be?

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(b)

Choose a representative and write out the elements of each of the cosets of $H .$ Then verify that each left coset is the same as the corresponding right coset. (This will verify that $H$ is normal.)

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(c)

Recall that we know the forms of all possible groups of the order you calculated in Task a. (Refer to your work from Discovery 13.6.) Determine which form $D_{n} / ⟨ r^{2} ⟩$ must take by determining whether it is cyclic or not.

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(d)

Based on Task c, is $D_{n} / ⟨ r^{2} ⟩$ Abelian? Is $D_{n}$ Abelian?

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Discovery 15.16.

In this activity, we will verify that equality (✶) in Fact 15.3.3 is independent of the choice of coset representatives $x$ and $y .$

Assume that $H$ is a subgroup of a group $G .$

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(a)

Prove that if $y_{1} H = y_{2} H,$ then also $(x y_{1}) H = (x y_{2}) H .$

Hint.

Apply Fact 15.3.1 to both the assumption and the desired conclusion. Don't forget to reverse order when you invert a product!

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(b)

Assume that $H$ is normal. Prove that if $x_{1} H = x_{2} H,$ then also $(x_{1} y) H = (x_{2} y) H .$

Hint.

Apply Fact 15.3.1 to both the assumption and the desired conclusion. Don't forget to reverse order when you invert a product!

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Subsection Commutators

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Discovery 15.17.

In this activity we will compute the commutator elements in $D_{n}$ for $n$ even.

Recall that every element of $D_{n}$ is of one of the following two forms,

r^{m}, r^{m} s,

for some exponent $m$ between $0$ and $n - 1,$ inclusive. Also recall the algebra rules in $D_{n} :$

\begin{aligned} r^{n} & = e, & s^{2} & = e, & s r^{k} = r^{n - k} s . \end{aligned}

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(a)

Simplify each of the four basic commutator element types below into one of the two forms of $D_{n}$ elements above.

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(i)

$(r^{m}) (r^{p}) {(r^{m})}^{- 1} {(r^{p})}^{- 1}$

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(ii)

$(r^{m} s) (r^{p}) {(r^{m} s)}^{- 1} {(r^{p})}^{- 1}$

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(iii)

$(r^{m}) (r^{p} s) {(r^{m})}^{- 1} {(r^{p} s)}^{- 1}$

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(iv)

$(r^{m} s) (r^{p} s) {(r^{m} s)}^{- 1} {(r^{p} s)}^{- 1}$

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(b)

From your computations, describe which of the $2 n$ elements of $D_{n}$ are commutator elements.

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Discovery 15.18.

Verify that the identity $e$ is always a commutator element of the group in which it resides.

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Discovery 15.19.

Verify that an inverse of a commutator element is also a commutator element. (Careful: When you invert a product you must reverse the order of multiplication!)

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Discovery 15.20.

We know $[G, G]$ is always a subgroup because we have defined it in terms of a generating set (the commutator elements). In this activity we will verify that it is normal.

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(a)

By inserting some $g g^{- 1}$ factors (which is just $e,$ after all) and rearranging brackets, verify that if we conjugate a commutator element,

g^{- 1} (x y x^{- 1} y^{- 1}) g,

the result is also a commutator element.

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(b)

Use the same sort of trick as in Task a to verify that a conjugate of a product of any (finite) number of commutator elements is also a product of commutator elements.

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(c)

Combine Task b with Discovery 15.19 and the definition of $[G, G]$ as generated by commutator elements to conclude that $[G, G]$ satisfies Condition 2 of Fact 15.3.5.

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Discovery 15.21.

In this activity we will verify that $G / [G, G]$ is always Abelian. To do this, we must verify that coset products

x [G, G] \cdot y [G, G] and y [G, G] \cdot x [G, G]

always have the same result.

Use equality (✶) in Fact 15.3.3 and Fact 15.3.1 to do so.

(Remember: When you invert a product you must reverse the order of multiplication!)

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Discovery 15.22.

In this activity we will verify that if $K$ is a normal subgroup of $G$ so that $G / K$ is Abelian, then $K$ must contain all of $[G, G] .$ As $K$ is assumed to be a subgroup, it is closed under products and inverses, so it suffices to show that $K$ contains each of the generating elements of $[G, G]$ (i.e. the commutator elements).

So consider commutator $x y x^{- 1} y^{- 1} .$ The assumption that $G / K$ is Abelian means that coset products

x^{- 1} K \cdot y^{- 1} K and y^{- 1} K \cdot x^{- 1} K

have the same result. (There is a reason those inverses are in there, hang tight ….)

Now apply equality (✶) in Fact 15.3.3 and Fact 15.3.1 to arrive at the conclusion that $x y x^{- 1} y^{- 1}$ must be in $K .$

(Remember: When you invert a product you must reverse the order of multiplication!)