02 - Groups

We present the formal idea of a group and give a few elementary examples. We also use these examples to sketch the idea of a group representation.

A group is a set $G$ together with a binary operation:

$$\star : G\times G \rightarrow G,$$

that is subject to three requirements. First, there is an identity element in $G$, say $1$, such that for all $g$ in $G$,

$$1\star g = g\star 1 = g.$$

Relatedly, every $g$ in $G$ must have an inverse element $g^{-1}$, such that

$$g\star g^{-1} = g^{-1}\star g = 1.$$

Finally, the operation $\star$ is associative, which means that for all $f,g,h$ in $G$

$$(f\star g)\star h = f\star(g\star h) = f\star g \star h.$$

Essentially, associativity means that the product of group elements

$$f\star g \star h \star \dots$$

is independent of the order of evaluation.

Quintessential examples of groups include the integers ($\mathbb{Z}$) with addition, or the real $(\mathbb{R})$ and complex $(\mathbb{C})$ numbers with either addition or multiplication. These examples immediately extent to the rational numbers:

$$\mathbb{Q} = \left\{ \frac{a}{b} \;\Big|\; a,b \in \mathbb{Z}\right\}.$$

We will now consider a few more examples of groups and their associated representations.

Finite Group Representations

Consider the cyclic groups:

$$\mathbb{Z}_{n} = \left\{ q\; \mathrm{mod} \;n \;\Big|\;q \in \mathbb{Z}\right\},$$

where $n$ is a positive number. Here $q$ mod or ``modulo'' $n$ means the remainder of the fraction $q/n$.

Any complex vector space $V$ can be a module for $\mathbb{Z}_{n}$, where the representation of $\mathbb{Z}_{n}$ is furnished by the set of scalars 

$$ \{ \pi_{q} = e^{2\pi i q/n} \;\Big|\; q \in \mathbb{Z}_{n}\}.$$

Here the scalars $\pi_{q}$ act as linear operators on $V$. To connect with our schematic picture from last time,

$$\pi : \mathcal{A} \rightarrow \mathsf{End}(V),$$

$\mathcal{A} = \mathbb{Z}_{n}$. The representation $\pi$ then takes any $q$ in $\mathbb{Z}_{n}$ to the endomorphism

$$\pi(q) = e^{2\pi i q/n}\mathbb{1},$$

where $\mathbb{1}$ is the identity map:

$$\mathbb{1}\cdot v = v,$$

for all $v$ in $V$.

A notable example that we will see a lot of is $\mathbb{Z}_{2} = \left\{\pm 1\right\}$. Notice that any real vector space can also serve as a module for $\mathbb{Z}_{2}$.

Abelian Groups

Abelian groups are those whose group operation is commutative. Often we refer to it as addition and use the associated notation, $a + b.$

In this context the unit element is typically written as $0$ and inverse elements $a^{-1}$ are often written as  $-a$.

The cyclic groups were abelian, and this addition notation can be inferred by the product of exponentials:

$$g_{1}\cdot g_{2} \rightsquigarrow e^{i\frac{m_{1}}{p}}e^{i\frac{m_{1}}{p}} = e^{i\frac{m_{1} + m_{2}}{p}},$$

for $g_{1},g_{2}$ in $\mathbb{Z}_{p}$.

Another common example that also happens to generalize the cyclic groups is the group $\mathsf{U}(1)$:

$$\mathsf{U}(1) = \left\{e^{i \theta} \;\Big|\; 0 \leq \theta < 2\pi \right\},$$

which also can be thought of as the unit circle

$$\mathsf{U}(1) = S^{1} =  \left\{ z \in \mathbb{C} \;\Big|\; |z|^{2} = 1\right\}.$$

Again, any complex vector space $V$ can serve as a module for $\mathsf{U}(1)$, where the representation acts by scalar multiplication. For some $v$ in $V$,

$$v\mapsto \pi(\theta)\cdot v = e^{i\theta}v.$$

Evidently any of the cyclic groups can be found inside $\mathsf{U}(1)$. More precisely, they are subgroups. A subgroup is a subset of a group that is also a group under the same group operation. Any subgroup must contain the same identity element from the original group, and therefore inverse elements of a subgroup also coincide with those in the original group. For semantic reasons a group is sometimes trivially considered as a subgroup of itself.

Infinite groups like $\mathsf{U}(1)$ that have a geometric interpretation are often called Lie groups. We’ll talk about them next time.