Normality in Abstract Algebra
students, imagine a club where every member can be moved around by the rules of the club and still the same “shape” remains. In Abstract Algebra, that idea is called normality. It is one of the key ideas behind normal subgroups and the construction of quotient groups. 🔑
By the end of this lesson, you should be able to:
- explain what normality means in a group,
- test whether a subgroup is normal,
- use conjugation to understand why normality matters,
- connect normality to quotient groups,
- recognize common examples of normal and non-normal subgroups.
Normality is important because it tells us when a subgroup fits perfectly into the group structure so that we can “divide out” by it and build a new group.
What Normality Means
Let $G$ be a group and let $H$ be a subgroup of $G$. The subgroup $H$ is called normal in $G$ if it stays the same under conjugation by any element of $G$.
That means for every $g \in G$,
$$gHg^{-1} = H.$$
Here, $gHg^{-1}$ means the set of all elements of the form $ghg^{-1}$ where $h \in H$.
This condition may look technical, but the idea is simple: if you “shuffle” the subgroup by an element of the larger group and then shuffle back, you land in the same subgroup. In other words, the subgroup is perfectly compatible with the whole group structure.
Another equivalent way to say this is:
$$gH = Hg \quad \text{for all } g \in G.$$
So a normal subgroup has the same left cosets and right cosets. This equality is a major sign that quotient groups can be formed.
Why conjugation matters
Conjugation is the operation
$$h \mapsto ghg^{-1}.$$
This is important because it shows how group elements act on each other inside the group. If conjugation keeps every element of $H$ inside $H$, then $H$ is stable under the group’s internal symmetry.
For example, in a symmetry group, conjugation can represent changing the coordinate system or relabeling the objects. A normal subgroup is one that does not depend on the viewpoint.
Ways to Recognize Normality
There are several standard ways to check whether $H$ is normal in $G$.
1. Direct conjugation test
Check whether
$$gHg^{-1} = H$$
for every $g \in G$.
This is the most direct definition, but it can be hard to verify if the group is large.
2. Coset test
Check whether left and right cosets are equal:
$$gH = Hg \quad \text{for all } g \in G.$$
If this holds, then $H$ is normal.
3. Kernel test
If $H$ is the kernel of a homomorphism $\varphi : G \to K$, then $H$ is normal in $G$.
That is,
$$\ker(\varphi) \trianglelefteq G.$$
This is one of the most useful facts in group theory. It shows that normal subgroups naturally appear as kernels of homomorphisms.
4. Special cases
Some subgroups are automatically normal:
- the trivial subgroup $\{e\}$,
- the whole group $G$,
- any subgroup of an abelian group.
Why? If $G$ is abelian, then for all $g,h \in G$,
$$ghg^{-1} = h,$$
because $gh = hg$. So every subgroup is normal in an abelian group.
Examples of Normal and Non-Normal Subgroups
Example 1: A normal subgroup in $\mathbb{Z}$
Consider the group $\mathbb{Z}$ under addition. Every subgroup has the form $n\mathbb{Z}$ for some integer $n \ge 0$.
Because $\mathbb{Z}$ is abelian, every subgroup is normal. So $3\mathbb{Z}$ is normal in $\mathbb{Z}$.
This is a great example because it shows that normality is automatic in abelian groups. The quotient group $\mathbb{Z}/3\mathbb{Z}$ is the set of congruence classes modulo $3$.
Example 2: A normal subgroup in $S_3$
Let $S_3$ be the symmetric group of all permutations of three objects. It has a subgroup
$$A_3 = \{e, (123), (132)\}.$$
This subgroup is normal in $S_3$ because it has index $2$. In general, every subgroup of index $2$ is normal.
Why? If there are only two cosets, then the left cosets and right cosets must match automatically.
This subgroup plays an important role because the quotient group $S_3/A_3$ has order $2$.
Example 3: A non-normal subgroup in $S_3$
Consider the subgroup
$$H = \{e, (12)\}.$$
This is a subgroup of $S_3$, but it is not normal. To see this, use conjugation by $(123)$:
$$ (123)(12)(123)^{-1} = (23). $$
Since $(23) \notin H$, we have
$$ (123)H(123)^{-1} \ne H. $$
So $H$ is not normal.
This example shows that being a subgroup is not enough. A subgroup must satisfy the normality condition to support quotient group construction.
Normality and Quotient Groups
Normality is the exact condition needed to form a quotient group.
If $H$ is normal in $G$, then the set of left cosets $G/H$ becomes a group with operation
$$ (gH)(kH) = (gk)H. $$
This operation is well-defined only when $H$ is normal. That means the result does not depend on which representatives from the cosets we choose.
For students, this is a key idea: quotient groups let us compress a group by “squashing” a normal subgroup into the identity.
Why well-definedness matters
Suppose $gH = g'H$ and $kH = k'H$. Then we want
$$ (gk)H = (g'k')H.$$
This is not automatically true for an arbitrary subgroup. The normality condition guarantees it.
Without normality, the multiplication rule could give different answers depending on the chosen representatives, which would make the quotient structure invalid.
A concrete quotient example
Take $G = \mathbb{Z}$ and $H = 4\mathbb{Z}$. Then the quotient group $\mathbb{Z}/4\mathbb{Z}$ has four elements:
$$0 + 4\mathbb{Z},\ 1 + 4\mathbb{Z},\ 2 + 4\mathbb{Z},\ 3 + 4\mathbb{Z}. $$
Addition works by adding representatives and then reducing modulo $4$.
This is the group version of modular arithmetic, and it depends on the fact that $4\mathbb{Z}$ is normal.
The Big Idea Behind Normality
Normality is not just a technical condition. It captures the idea that a subgroup behaves the same from every group-theoretic viewpoint.
Here is a helpful way to think about it:
- a subgroup is a smaller group inside a bigger group,
- a normal subgroup is one that fits so smoothly into the bigger group that the whole group can be simplified by collapsing it,
- quotient groups record the structure that remains after that collapse.
This is why normality is central in group theory. It is the bridge between studying a group directly and studying its simplified versions.
Common Facts to Remember
A few important results are used often:
- If $H \trianglelefteq G$, then the set of cosets $G/H$ forms a group.
- If $\varphi : G \to K$ is a homomorphism, then $\ker(\varphi)$ is normal in $G$.
- Every subgroup of an abelian group is normal.
- Every subgroup of index $2$ is normal.
- If $H$ is normal in $G$ and $K$ is normal in $H$, that does not automatically mean $K$ is normal in $G$.
That last point is important because normality does not always pass upward through subgroups.
Conclusion
Normality is the property that makes a subgroup compatible with the whole group under conjugation. students, you can test normality by checking conjugates, comparing left and right cosets, or recognizing a subgroup as a kernel. Normal subgroups are essential because they are exactly the subgroups that allow quotient groups to be formed.
In Abstract Algebra, normality is a foundational idea. It explains when a group can be simplified without losing the ability to keep group operations well-defined. Once you understand normality, quotient groups become much easier to understand, and many important examples in algebra start to make sense. 🌟
Study Notes
- A subgroup $H$ of a group $G$ is normal if $gHg^{-1} = H$ for every $g \in G$.
- Equivalent condition: $gH = Hg$ for all $g \in G$.
- Normality means the subgroup is stable under conjugation.
- The kernel of any group homomorphism is a normal subgroup.
- Every subgroup of an abelian group is normal.
- Every subgroup of index $2$ is normal.
- Normal subgroups are exactly the subgroups used to form quotient groups.
- The quotient group $G/H$ is well-defined only when $H$ is normal.
- In $S_3$, the subgroup $A_3 = \{e,(123),(132)\}$ is normal.
- In $S_3$, the subgroup $\{e,(12)\}$ is not normal because conjugation can move it outside itself.
- Normality is a central idea connecting subgroups, homomorphisms, and quotient groups.