UA-AUG-AUMAT229 Normal subgroups and quotient groups: Pre-read

Section 15.3 Normal subgroups and quotient groups: Pre-read

Subsection Recognizing when two cosets are actually the same coset

It will be often the case that we want a simple way to check whether cosets

x H, y H

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are actually the same coset without listing out all the elements of both cosets. Recall that (left) cosets partition the group, so each element is in exactly one coset. As

H

is a subgroup, it contains

e,

hence

y H

always contains

y .

Therefore, if we knew that

x H

contained

y

as well, then

x H

would have to be the same coset as

y H .

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Fact 15.3.1.

If $H$ contains $y^{- 1} x,$ then $x H = y H .$

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Subsection Normal subgroups and the quotient group

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In the textbook, you read about normal subgroups. The normal property is precisely the condition needed so that defining a group operation on cosets makes sense.

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Definition 15.3.2. Normal subgroup.

A subgroup that is made up of one or more whole conjugacy classes taken together.

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Fact 15.3.3.

Recall that we have defined multiplication on the quotient $G / H$ by taking

x H \cdot y H

to mean the collection of all results of computing a product of an element in coset $x H$ times an element in coset $y H$ (in that order). If $H$ is normal, the result of such a product of cosets will always be again a coset. In fact, we will always have

\begin{matrix} (✶) & x H \cdot y H = (x y) H . \end{matrix}

Furthermore, this operation makes $G / H$ into a group.

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Warning 15.3.4.

When $H$ is normal, then $G / H$ is a group, but it is not a subgroup of $G$ .

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The way you should think of

G / H

is that it is a new group that shares something of the same algebraic patterns as

G

(since the group operation on

G / H

involves using the group operation of

G

on pairs of elements from cosets of

H

), but with the algebraic patterns of

H

“removed” or “divided out.”

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Here are some other ways of checking whether a group is normal.

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Fact 15.3.5.

Suppose $H$ is a subgroup of a group $G .$ Then $H$ is normal if any one of the following conditions are true.

Whenever $H$ contains a group element $h,$ it must contain the entire conjugacy class of $h$ (as a class in $G$ ).
For each $h$ in $H,$ the result of $g^{- 1} h g$ is also in $H$ for every element $g$ of $G .$
(Only for $G$ finite.) For each $h$ in $H,$ the result of $x^{- 1} h x$ is also in $H$ for every element $x$ in some set of generators for $G .$
(For $G$ finite or infinite.) For each $h$ in $H,$ the result of both $x^{- 1} h x$ and $x h x^{- 1}$ is also in $H$ for every element $x$ in some set of generators for $G .$
For each element $g$ of $G,$ the left coset $g H$ is the same collection of group elements as the right coset $H g .$

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Note 15.3.6.

If one of the conditions in Fact 15.3.5 is true, then they all will be true, but you only need to check one.

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Subsection Commutators

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Abelian groups are obviously simpler than non-Abelian groups. We will see in an example in the Discovery guide following this Pre-read section that it is possible for a non-Abelian group to have an Abelian quotient. (See Discovery 15.15.) This is desirable because the quotient still retains some of the information of the group

G,

but in an Abelian form.

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The natural question is: for non-Abelian

G,

what is the smallest possible normal subgroup

H

so that the quotient

G / H

is Abelian? (The reason we want

H

to be as small as possible is so that

G / H

will be as big as possible, hence will retain more of the information of

G .

)

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G

were Abelian, then

x y = y x

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would be true for every pair of elements

x, y

G .

Rearranging, if

G

were Abelian, then

x y x^{- 1} y^{- 1} = e

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would always be true. (Remember: Order of multiplication matters in a non-Abelian group!)

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Definition 15.3.7. Commutator element.

A group element that is somehow of the form $x y x^{- 1} y^{- 1}$ for some pair of group elements $x, y .$

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We want to form a subgroup

H

that will allow us to “divide away” all the troublesome elements in the quotient

G / H .

And here the commutators are precisely the troublesome elements —

G

is non-Abelian precisely because there are non-trivial commutators. The smallest subgroup that contains a particular collection of elements is the one generated by those elements.

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Definition 15.3.8. Commutator subgroup.

The subgroup generated by the collection of commutator elements. We write $[G, G]$ to denote this subgroup.

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Warning 15.3.9.

In general, the commutator subgroup contains more than just the commutator elements. (Look back at the definition of the subgroup generated by a collection of elements.)

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Fact 15.3.10.

The commutator subgroup is always normal, the quotient $G / [G, G]$ is always Abelian, and if any other normal subgroup $K$ of $G$ has the property that if $G / K$ is abelian, then $K$ contains $[G, G]$ as a subgroup.