Cosets and Normal Subgroups

1. The Quest for a Group of Cosets

Recall that for a subgroup $H\subseteq G$, the set of left cosets, denoted $G/H$, is the collection of sets $\{gH\mid g\in G\}$. We have seen that these cosets partition the group $G$. A natural and ambitious question to ask is whether this collection of sets can itself form a group. If it could, the elements of our new group would be the cosets themselves.

To start building intuition, let's explore a familiar case where this works beautifully: the integers under addition.

The example in the video suggests a natural way to define an operation on cosets. If our group operation is written multiplicatively, we might propose the following rule for "multiplying" two cosets $aH$ and $bH$:

This definition seems straightforward. It satisfies the group axioms of associativity, identity (the coset $eH=H$ acts as the identity), and inverses (the inverse of $gH$ is $g^{-1}H$), provided the operation itself is valid.

However, a subtle but profound problem lurks. A single coset can be represented by any of its elements. For instance, if $h_1\in H$, then the coset $aH$ is identical to the coset $(a{h}_1)H$. The problem of being well-defined is this: does our result depend on which representative we choose for the coset? If we choose a different representative, say $a^{\prime }=a{h}_1$ for $aH$, and $b^{\prime }=b{h}_2$ for $bH$, will the product $(a^{\prime }{b}^{\prime })H$ still be the same as $(ab)H$? If the answer is no, our proposed multiplication rule is ambiguous and therefore useless.

2. The Condition for a Well-Defined Operation

Let's investigate precisely what conditions the subgroup $H$ must satisfy for our coset multiplication to be well-defined. We'll use a concrete example where the rule fails, and then derive the general condition that prevents this failure.

The following video does an excellent job of first demonstrating the problem with a counterexample from the dihedral group, and then deriving the necessary condition from first principles.

As the video rigorously demonstrated, for the operation $(aH)(bH)=(ab)H$ to be well-defined, the subgroup $H$ must satisfy a special property. The derivation shows that for any element $g\in G$ and any element $h\in H$, the element $gh{g}^{-1}$ must also be an element of $H$.

This can be stated more compactly as requiring the set of all such conjugated elements to be a subset of $H$:

This is the fundamental condition we have been searching for. Subgroups that satisfy this property are given a special name.

3. Normal Subgroups

We now elevate this condition to a formal definition.

Definition: A subgroup $N\subseteq G$ is called a normal subgroup of $G$ if for every $g\in G$, the set $gN{g}^{-1}$ is equal to the set $N$. We denote this as $N◃G$.

The set $gN{g}^{-1}=\{gn{g}^{-1}\mid n\in N\}$ is called the conjugate subgroup of $N$ by $g$. A normal subgroup is therefore a subgroup that is equal to all of its conjugate subgroups—it is "invariant" under conjugation by any element of the parent group.

The following resource provides several equivalent and useful ways to think about this definition.

4. Normality and Conjugacy Classes

The condition $gn{g}^{-1}\in N$ for all $n\in N$ provides a powerful and intuitive link back to our previous lesson on conjugacy classes. It states that if you take any element $n$ from a normal subgroup $N$, all of its conjugates must also lie within $N$. This means that a normal subgroup cannot contain just part of a conjugacy class; if it contains one element from a class, it must contain the entire class.

This leads to a beautifully simple and profound characterization:

Theorem: A subgroup $H\subseteq G$ is normal if and only if it is a union of conjugacy classes of $G$.

This explains why some subgroups, like the subgroup of reflections in $D_4$, are not normal. As we saw in the previous lesson, the reflections $\{s,r^{2}s\}$ form one conjugacy class, and $\{rs,r^{3}s\}$ form another. A subgroup like $H=\{e,s\}$ contains the identity class $\{e\}$ but only part of the class $\{s,r^{2}s\}$. It "breaks" a conjugacy class and is therefore not normal.

Let's read a formal statement and proof of this connection.

5. Examples and Tests for Normality

With these definitions in hand, let's identify some important examples of normal subgroups.

• Trivial Normal Subgroups: For any group $G$, the subgroup containing only the identity, $\{e\}$, and the group $G$ itself are always normal.
• Subgroups of Abelian Groups: In an abelian group, $gn{g}^{-1}=g{g}^{-1}n=n$, so the condition is trivially satisfied. Therefore, all subgroups of an abelian group are normal. This explains why the cosets of $5\mathbb{Z}$ in $\mathbb{Z}$ formed a group.
• The Center of a Group: The center $Z(G)$ consists of all elements that commute with every element of $G$. For any $h\in Z(G)$, $gh{g}^{-1}=g{g}^{-1}h=h$, so $Z(G)$ is always a normal subgroup.
• Subgroups of Index 2: If a subgroup $H$ has index $[G:H]=2$, it means there are only two left cosets: $H$ and $gH$ for some $g\notin H$. There are also only two right cosets: $H$ and $Hg$. Since the cosets must partition the group, we must have $gH=G\setminus H$ and $Hg=G\setminus H$, which implies $gH=Hg$. Thus, any subgroup of index 2 is normal. A prime physical example is the alternating group $A_n$ (the group of even permutations), which is a subgroup of index 2 in the symmetric group $S_n$ and is therefore normal. This is crucial in the theory of permutations.
• A Physical Example: Consider the Rubik's Cube group. Let $C_0$ be the subgroup of moves that change the orientation of the individual corner and edge pieces but leave their positions fixed. This subgroup is normal. The physical reasoning is a perfect illustration of conjugation:
  1. Perform an arbitrary sequence of moves, $g$.
  2. Perform a move $n\in C_0$ that only twists some pieces in place.
  3. Undo the first sequence of moves, $g^{-1}$.
     The net result, $gn{g}^{-1}$, is a configuration where every block has returned to its original position, though its orientation may have changed. Thus, the resulting move is still in $C_0$. Since this holds for any $g$, $C_0$ is a normal subgroup.

Conclusion

In this lesson, we have identified the crucial property that allows the set of cosets to form a group. This exploration has led us to the definition and significance of normal subgroups, unifying the concepts of cosets and conjugacy classes.

Here are the key takeaways:

• The set of cosets $G/H$ can form a group under the operation $(aH)(bH)=(ab)H$ only if this operation is well-defined.
• The operation is well-defined if and only if the subgroup $H$ is normal.
• A subgroup $H$ is normal in $G$, denoted $H◃G$, if it satisfies any of these equivalent conditions:
  • It is invariant under conjugation: $gH{g}^{-1}=H$ for all $g\in G$.
  • Its left and right cosets are the same: $gH=Hg$ for all $g\in G$.
  • It is a union of conjugacy classes of $G$.
• We identified several key examples, including all subgroups of abelian groups, the center $Z(G)$, and any subgroup of index 2.

Having established the condition for their existence, we are now ready to formally construct and analyze these new groups built from cosets. In our next lesson, we will define the quotient group $G/N$ and explore how it reflects the structure of the original group $G$, providing a way to "factor out" the structure of the normal subgroup $N$. This will be an indispensable tool as we move towards understanding group homomorphisms and representations.