Cosets and Normal Subgroups in Abstract Algebra 📂Abstract Algebra

Cosets and Normal Subgroups in Abstract Algebra

Definition ¹

The set $G$ and its subgroup $H$ such that $aH = \left\{ ah \ | \ h \in H \right\}$ is called the Left Coset and $Ha = \left\{ ha \ | \ h \in H \right\}$ is called the Right Coset. Here, $a \in G$ and $aH, Ha \subset G$.
The number of left (right) cosets of $H \leqslant G$ is denoted as $(G : H)$ and called the Index of $H$ in $G$.
If $H$ is a subgroup of $G$ and for all $g \in G$, $gH = Hg$ holds, then $H$ is called a Normal Subgroup of $G$, denoted as $H \triangleleft G$.
$G$ is called Simple if it doesn’t have a $H \triangleleft G$ other than $H = \left\{ e \right\}$ or $H = G$. In other words, $G$ is simple if it only has $\left\{ e \right\}$ and itself as normal subgroups.

Explanation

Cosets

The idea of cosets inevitably leads algebra to a higher level.

For example, the set $3 \mathbb{Z} = \left\{ \cdots, -6, -3, 0 , 3, 6 , \cdots\right\}$ of multiples of $3$ forms a group, and especially since $\mathbb{Z}$ is an Abelian group, $3 \mathbb{Z} \triangleleft \mathbb{Z}$ holds.

If we think about adding integers here, $$ \begin{align*} 1 + 3 \mathbb{Z} =& \left\{ \cdots, -5, -2, 1 , 4, 7 , \cdots\right\} \\ 2 + 3 \mathbb{Z} =& \left\{ \cdots, -4, -1, 2 , 5, 8 , \cdots\right\} \\ 3 + 3 \mathbb{Z} =& \left\{ \cdots, -3, 0 , 3, 6 , 9 , \cdots\right\} = 3 \mathbb{Z} \\ 4 + 3 \mathbb{Z} =& \left\{ \cdots, -2, 1 , 4, 7 , 10 , \cdots\right\} = 1 + 3 \mathbb{Z} \\ 5 + 3 \mathbb{Z} =& \left\{ \cdots, -1, 2 , 5, 8 , 11 , \cdots\right\} = 2 + 3 \mathbb{Z} \end{align*} $$ it resembles adding integers in $\pmod{3}$.

This means we can think of a new group consisting of sets as elements, like $\mathbb{Z}_{3} : = \left\{ 3 \mathbb{Z} , 1 + 3 \mathbb{Z} , 2 + 3 \mathbb{Z}\right\}$. Such a newly formed group is called a Quotient group. It’s a concept that’s quite difficult to understand at first, usually due to a misunderstanding of cosets. Don’t underestimate it as if it’s trivial and unused; it’s crucial to master cosets by writing them down to ease understanding of later topics.

Index

It says left and right without distinguishing because there’s no particular need to. Originally, an index is defined by the number of left cosets, but in fact, since there exists a one-to-one correspondence with right cosets, it can also be defined by the number of right cosets.²

Normality

Checking whether $gH$ and $Hg$ form a group and whether $gH = Hg$ holds reminds one of the definition of continuity learned in courses. Given the term Normal, it suggests a very strong condition and many useful properties can be inferred.

Immediately known from the definition is that for the identity element $e$ of $G$, $\left\{ e \right\} \triangleleft G$ exists. A bit of thought reveals that for an Abelian group $G$, if $H \leqslant G$, then approximately $H \triangleleft G$ exists.

Simplicity

For example, for a prime number $p$, $\mathbb{Z}_{p}$ doesn’t have any subgroups other than the trivial group or itself, thus it forms a simple group.