Groups, Homomorphisms, and Group Actions

Note

Video: Groups, Homomorphisms, and Group Actions — 2:44:50

This section corresponds to 2:44:50 in Proof of Fermat Last Theorem FROM SCRATCH.

Groups

Before defining a group, we first need to clarify one basic concept first.

📌 Binary operation

A binary operation \(*\) on a set \(S\) is a function that assigns to each ordered pair \((a,b)\) of elements of \(S\) some element of \(S\).

Associativity: \((a*b)*c = a * (b* c)\)
Commutativity: \(a*b = b*a\)

The most familiar binary operations we already know would definitely be addition and multiplication. For example, addition can be viewed as an operation that assigns the value 7 to the pair \((3, 4)\). Likewise, multiplication can be seen as an operation that assigns the value of 20 to the pair \((4, 5)\).

However, we may want to define more unusual operations as well.

📌 Group

A group \((G, *)\) is a set \(Gf\) with a binary operation \(*\) satisfying three axioms:

The operation \(*\) is associative: \(\forall a,b,c \in G:\ (a*b)*c=a*(b*c)\)
There is an element \(e\in G\) such that \(e*x = a * e = a\) for all \(a \in G\). This is called the identity element.
For each \(a\in G\), there is an element \(a^{-1}\in G\), such that \(a*a^{-1}=a^{-1}* a=e\). This is called an inverse element.

A group is abelian if its operation is also commutative.

The number of elements in a gorup \(G\) is called its order, denoted by \(|G|\).

🏷️ Examples

The set of integers with addition is a group:

The addition operation is clearly an operation on \(\mathbb{Z}\), as it always returns another integer.
Addition is associative.
The identity element is zero.
Each element \(a\in\mathbb{Z}\) has an inverse, namely \(-a\).

It is also an abelian group, since addition is commutative.

Similarly, \((\mathbb{Q}, +)\), \((\mathbb{R}, +)\), and \((\mathbb{C}, +)\) are also groups.

🏷️ Counter example: \((\mathbb{Q}, *)\)

This is not a group, as there is no inverse element for zero: there is no such element for which we can write \(0*b=b*0=1\).

So the same set can behave very differently, depending on what operations we put on it!

🏷️ Example: \((\mathbb{Z}_n, +)\)

The group \((\mathbb{Z}_n, +)\). Remember that \(\mathbb{Z}_n\) is a subset \(\mathbb{Z}_n=\{0,1,2\cdots, n-1\}\).

Note that the addition operation is defined differently here: \(a + b\ (\text{mod}\ n)\)

Take for example \((\mathbb{Z}_3, +)\)

\[ \begin{array}{l} 1 + 0 = 1 \\ 1 + 1 = 2 \\ 1 + 2 = 0 \end{array} \]

Let’s verify that \((\mathbb{Z}_5, +)\) is a group:

The \(+\)-operation is associative
There is a zero element: \(0 + a = a + 0 =a\)
Every element has an inverse:

\[ \begin{array}{l} 1 + 4 = 4 + 1 = 0\ (\text{mod}\ 5) \\ 2 + 3 = 3 + 2 = 0\ (\text{mod}\ 5) \\ \vdots \end{array} \]

Similarly, \((\mathbb{Z}_n, +)\) is a group.

🏷️ Example: \((\mathbb{Z}_n \times \mathbb{Z}_m, +)\)

The addition operation is defined as

\[ (a_1, b_1) + (a_2, b_2) := ((a_1 +b_1)\ \text{mod}\ n, (a_2 + b_2) \ \text{mod}\ m) \]

With this operation, it forms a group. Let’s check this for \((\mathbb{Z}_3 \times \mathbb{Z}_4, +)\):

With the above addition operation, we always get another element in the group: \[ \begin{array}{l} 1 + 2 = 1 + 3 = (2, 1) \in \mathbb{Z}_3 \times \mathbb{Z}_4 \\ 2 + 3 = 3 + 2 = (1, 1) \in \mathbb{Z}_3 \times \mathbb{Z}_4 \end{array} \]
The zero element is \((0, 0)\)
For every element, there is an inverse, for example \((2, 2) + (1, 2) = (0, 0)\).

So indeed, \((\mathbb{Z}_n \times \mathbb{Z}_m, +)\) is a group.

❗ More generally, we have that

\[ (\mathbb{Z}_{n_1} \times \mathbb{Z}_{n_2} \times \mathbb{Z}_{n_3}\cdots, +) \]

forms a group.

🏷️ Example: \((GL_n(\mathbb{R}), *)\)

The collection of invertible matrices \(GL_n(\mathbb{R})\) with matrix multiplication froms a group:

\(\forall A, B\in GL_n(\mathbb{R}): A*B \in GL_n(\mathbb{R})\) as \(\det(AB)=\det(A)\det(B)\neq0\).
Matrix multiplication is associative
Obvisouly, \(GL_n(\mathbb{R})\) contains a zero element.
As the determinant is non-zero by definition, every matrix has an inverse matrix.

🏷️ Example: \(n\)-th roots of unity

If we define \(\mu_n = \{z\in\mathbb{C}\ |\ z^n=1 \}\) with multiplication, \((\mu_n, *)\) forms a group.

\(\forall z_1, z_2 \in \mu_n: (z_1z_2)^n = z_1^nz_2^n =1\)
Multiplication is associative
There is one identiy element, namely \(1^n = 1\)
Every element has an inverse, as \(1 / z^n\) is also in the group.

Note

🚨 When the operation is obvious, we usually just state the set without explicitly mentioning its operation!

🧪 Rubik’s cube

The moves for the Rubik’s cube form a group as well: \([R, L, F, B, U, D]\).

📌 Subgroup

A subset of \(H\) of a group \(G\) is a subgroup (\(H\leq G\)) if \(H\) is itself a group under the operations of \(G\).

It is a concept that is very similar to a subspace or subset, so it is basically a smaller group sitting inside a larger group.

🏷️ Examples

\[ (\mathbb{Z}, +) \leq (\mathbb{Q}_n, +) \]

Also,

\[ (SL_n(\mathbb{R}), *) \leq (GL_n(\mathbb{R}), *) \]

Another example would be \((\mu_2, *) \leq (\mu_4, *)\).

📌 Cyclic groups

A group \(G\) is cyclic if it can be generated by a single element. We write \(G=\langle a \rangle = \{a^n\ |\ n \in \mathbb{Z}\}\). The element \(a\) is a generator of \(G\).

So the whole group can be built by applying the generator multiple times on itself:

\[ \begin{array}{c} a \\ a * a \\ a * a * a \\ a * a * a * a \\ \vdots \end{array} \]

🏷️ Examples

\[ \mathbb{Z} = \langle 1 \rangle = \langle -1 \rangle \]

Also

\[ \mathbb{Z}_n = \langle 1 \rangle = \langle -1 \rangle \]

Note that we also have

\[ \mathbb{Z}_7 = \langle 1 \rangle = \langle -1 \rangle = \langle 3 \rangle \]

as by adding 3, we get

\[ \{3, 6, 2, 5, 1, 4, 0\} = \mathbb{Z}_7 \]

Another example of a cyclic group is

\[ \mu_n=\langle e^{\frac{2\pi i}{n}}\rangle \]

📌 Isomorphic groups

So, to understand the group, we need to understand how the elements interact under that operation. One way to do that is to make an operation table.

For example, let’s look at the operation tables of both \(\mathbb{Z}_3\)

\[\begin{array}{|c|c|c|c|} + & 0 & 1 & 2\\ \hline 0 & 0 & 1 & 2\\ \hline 1 & 1 & 2 & 0\\ \hline 2 & 2 & 0 & 1 \end{array}\]

and \(\mu_3\):

\[ \begin{array}{|c|c|c|c|} + & 1 & e^{\frac{2\pi i}{3}} & e^{\frac{4\pi i}{3}}\\ \hline 1 & 1 & e^{\frac{2\pi i}{3}} & e^{\frac{4\pi i}{3}}\\ \hline e^{\frac{2\pi i}{3}} & e^{\frac{2\pi i}{3}} & e^{\frac{4\pi i}{3}} & 1\\ \hline e^{\frac{4\pi i}{3}} & e^{\frac{4\pi i}{3}} & 1 & e^{\frac{2\pi i}{3}} \end{array} \]

Upon closer inspection, we notice that these operation tables have exactly the same structure, as there turns out to be an obvious one-to-one correspondence between the elements of each group:

\[ \begin{array}{c} 0 \longleftrightarrow 1 \\ 1 \longleftrightarrow e^{\frac{2\pi i}{3}} \\ 2 \longleftrightarrow e^{\frac{4\pi i}{3}} \\ \end{array} \]

👉 Two groups \((G, *)\) and \((G', *')\) are isomorphic if their elements can be paired up in a way that preserves the group’s structural properties. We write \(G\cong G'\).

❗ The two groups have very little in common. One group consists of integers, the other contains exponents with even imaginary numbers. Also, the first group is a group under addition, whereas the second group is a group under multiplication. However, although the two groups are very different, they have exactly the same structure!

🧠 What matters is how the elements interact with each other. If two groups have the same operation structure after relabelling the elements, they are called isomorphic.

👉 \(\mathbb{Z}_3 \cong \mu_3\)

📌 Isomorphism: the map

An Isomorphism \(\phi\) from a group \((G, *)\) to a group \((G', *')\) is a function \(\phi: G\rightarrow G'\) that satisfies two conditions:

\(\phi\) is a bijection (one-to-one and onto).
\(\phi\) preserves the operation: For all \(a,b\in G\), \[ \phi(A*B)=\phi(a)*'\phi(b) \]

An automorphism is an isomorphism from a group \(G\) to itself.