MTH3003 Lecture 7

Following on from previous lectures, we can continue building on our knowledge of subgroups by exploring normal subgroups, cosets, simplicity, and a recap of Lagrange’s Theorem.

Cosets

Let $H\le G$ and $g\in G$. A left coset of $H$ in $G$ is something of the form:

$$gH=\{gh:h\in H\}$$

There are also right cosets of the form $Hg$, but they behave similarly, just with different notation. The set of all cosets of $H$ in $G$ is denoted by $G/H$, so $G/H=\{gH:g\in G\}$.

Warning

In general, $G/H$ is not a group itself!

The number of distinct cosets of $H$ in $G$ is called the index of $H$ in $G$ and is denoted $[G:H]$.

We also know a few equivalent properties for groups $H\le G$ with $a,b\in G$:

1. $a^{-1}b\in H$
2. $aH=bH$
3. $a\in bH$
4. $a=bh$ for some $h\in H$

From this, we can establish the following corollary: for all $a,b\in G$, cosets are either disjoint ($aH\cap bH=\varnothing$) or equal ($aH=bH$). Every element of $G$ lies in a coset of $H$, and for all $g\in G$, $|H|=|gH|$.

Normal Subgroups

A subgroup $N$ of a group $G$ is normal if $g^{-1}kg\in N$ for all $k\in N$ and all $g\in G$. The notation we use is $N⊴G$.

There are a few equivalent ways to define this:

• $N⊴G$ means that for all $g\in G$, we have $g^{-1}Ng=N$.
• $N⊴G$ means that for all $g\in G$, we have $Ng=gN$.

Normal subgroups are incredibly useful because we can “divide” by them and still get a group, creating what are called quotient groups.

Examples of Normal Subgroups

1. $⟨e_G⟩⊴G$
2. $G⊴G$
3. If $G$ is Abelian and $H\le G$, then $H⊴G$. (All subgroups of an Abelian group are normal).

Warning

When checking if $H$ is normal in $G$, you must show that $g^{-1}hg\in H$ for every element $h\in H$ and every $g\in G$. Don’t fool yourself by accidentally assuming $G$ is Abelian!

Simplicity

Groups in which no proper nontrivial subgroup is normal are special, and play a large role in group theory.

A group $G$ is simple if its only normal subgroups are the trivial group $⟨e_G⟩$ and $G$ itself. For example, any cyclic group of prime order, $C_p$, is simple.

When working with normal subgroups, we often use the Order Switching Lemma. Let $N⊴G$. Suppose $g\in G$ and $k\in N$. Then there exist $k^{\prime },k^{\prime \prime }\in N$ such that:

$$gk=k^{\prime }g\text{and}kg=g{k}^{\prime \prime }$$

This lemma only works when one of the groups is normal!

Furthermore, if $H\le G$ and $N⊴G$, we can prove that $HN=NH$, $H\cap N⊴H$, and $HN\le G$.

Lagrange’s Theorem Recap

Let $H$ be a subgroup of a finite group $G$. Lagrange’s Theorem states that:

$$|G|=[G:H]\cdot |H|$$

In particular, the order of $H$ divides the order of $G$.

Cauchy's Theorem (Partial Converse to Lagrange)

If $G$ is a finite group and $p$ is a prime number dividing $|G|$, then $G$ contains a subgroup of order $p$.

---

Pre-Lecture Notes from University Notes

• Definitions and examples of left cosets ($gH$) and the set of all cosets $G/H$.
• Equivalent properties of cosets and the proof that cosets are either disjoint or equal.
• Definition of a normal subgroup ($N⊴G$) using $g^{-1}kg\in N$.
• Introduction to simple groups (groups whose only normal subgroups are trivial or the group itself).
• The Order Switching Lemma for manipulating elements in normal subgroups.
• Recap of Lagrange’s Theorem ($|G|=[G:H]\cdot |H|$) and an introduction to Cauchy’s Theorem.