Introduction to Groups

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1) What is a Group?

A group is a set of elements G = \{E, A_1, A_2, \ldots, A_n\} together with a binary operation (often called "multiplication") that satisfies four conditions:

1. Closure: For all A, B \in G, the product AB \in G.
2. Associativity: For all A, B, C \in G, (AB)C = A(BC).
3. Identity: There exists an element E \in G (also common - I \in G) such that EA = AE = A for all A \in G.
4. Inverses: For every A \in G, there exists A^{-1} \in G such that AA^{-1} = A^{-1}A = E.

The order of a group is the number of elements it contains, written |G|.

2) Examples of Types of Groups

There are many types of groups! Here are just some examples of important families, organized by whether they have finitely or infinitely many elements. In this course, we will start with finite groups (easier to code) and work up to (special types of) infinite groups.

Finite groups:

• Cyclic groups \mathbb{Z}_n — generated by a single element (e.g. rotations of a regular n-gon), order n. The integers (\mathbb{Z}, +) form the infinite cyclic group.
• Dihedral groups D_n — symmetries of a regular n-gon (rotations + reflections), order 2n.
• Symmetric (permutation) groups S_n — all permutations of n objects, order n!.
• Rubik's cube group — a subgroup of S_{48} with approximately 4.3 \times 10^{19} elements. Nathan Carter uses this example in Chapter 1 of his textbook Visual Group Theory.

Infinite (continuous) groups:

• Rotation groups SO(n) — the special orthogonal group of n \times n rotation matrices with the property R^T=R^{-1} such that R^TR=RR^T=I. SO(3) describes all rotations in 3D.
• Unitary groups SU(n) - the special unitary group, essentially the complex version of rotation matrices, U^\dagger = U^{-1} where \dagger is the complex conjugate transpose, such that U^\dagger U = UU^\dagger = I. SU(2) describes the transformations of spin \frac{1}{2} particles.
• Euclidean groups E(n) — rotations + translations in n dimensions.
• General linear group GL(n, \mathbb{R}) — all invertible n \times n real matrices.

Groups in science:

• Point groups — finite subgroups of O(3), used to classify molecular symmetries.
• Space groups — the 230 crystallographic groups (point symmetries + translations), fundamental in condensed matter physics and materials science.
• Standard Model of Particle Physics - the product group SU(3) \times SU(2) \times U(1) describing the internal symmetries of the strong, weak, and electromagnetic forces.

3) P(3) \cong S_3 \cong D_3: A Key Example

To begin our exploration of properties of groups, we will start with the small, yet nontrivial group P(3). P(3) is the group of all permutations of 3 objects. There are 3! = 6 ways to rearrange three things, so P(3) has 6 elements: \{E, A, B, C, D, F\}. We label these elements to match the conventions used in Dresselhaus 1.1.

There are three types of elements or conjugacy classes for the permutations of three objects (labeled 1, 2, 3).

• E = (123) — the identity, everything stays put
• A = (132), B = (321), C = (213) — swaps (two objects exchange, one stays)
• D = (312), F = (231) — cycles (all three objects move): D sends 1 \to 3 \to 2 \to 1, F sends 1 \to 2 \to 3 \to 1

If we instead think of these permutations as symmetries of a triangle in 2D (the dihedral group D_{3}), the swaps correspond to reflections (mirrors through each vertex) and the cycles correspond to rotations (120° and 240°).

Each element can also be represented as a matrix — either a 3 \times 3 permutation matrix acting on 3 objects, or a 2 \times 2 rotation/reflection matrix acting on the plane. The key property of a representation D(g) for g \in G is that matrix multiplication mirrors group multiplication: D(g \cdot h) = D(g) \cdot D(h).

4) Multiplication Tables

Rather than describe groups by their representations (for finite groups), we can use their multiplication table (Dresselhaus 1.2). In fact, you can even build up multiplication tables without prior knowledge of what the group does, using the rearrangement theorem, closure, and associativity.

The interactive explorer below shows the multiplication table for P(3). Click any cell to see the composition as an action on the triangle, or click a header to apply a single element.

5) Properties of Groups

The interactive explorer below highlights key group properties directly in the P(3) multiplication table. Select a property tab to explore it.

5.1) Rearrangement Theorem

If \{E, A_1, \ldots, A_n\} = G, then \{A_k E, A_k A_1, \ldots, A_k A_n\} = G for any fixed A_k.

Proof: If A_r = A_k^{-1} X, then A_k A_r = A_k A_k^{-1} X = X (every element is contained). If X appears twice, i.e. X = A_k A_r = A_k A_s, then A_k^{-1} A_k A_r = A_k^{-1} A_k A_s, meaning A_r = A_s, a contradiction on the definition of a group.

\Rightarrow Every row and column of a multiplication table contains each element once and only once.

This leads to a powerful identity: for any function f on the group,

5.2) Inverses

For each element g, there is exactly one element g^{-1} such that g \cdot g^{-1} = E. In the multiplication table, E appears exactly once in each row and column — the column where it appears in row g tells you g^{-1}.

For P(3): E^{-1} = E, A^{-1} = A, B^{-1} = B, C^{-1} = C, D^{-1} = F, F^{-1} = D.

Note that the mirrors are each their own inverse!

5.3) Subgroups

A subgroup H \subseteq G is a subset of elements that also forms a group under the same operation. In the multiplication table, a subgroup appears as a closed block — multiplying any two elements of the subgroup gives another element of the subgroup.

The subgroups of P(3):

• \{E, D, F\} — the cyclic group C_3 (3-fold rotations), order 3
• \{E, A\}, \{E, B\}, \{E, C\} — order-2 subgroups (each mirror is its own inverse)

Key observations:

• All subgroups contain the identity.
• Each subgroup still satisfies the rearrangement theorem.

5.4) Conjugacy Classes

An element B is conjugate to A if B = X A X^{-1} for some X \in G. This is a "change of basis" for our elements.

All elements that can be obtained by conjugation of a given group element form a class (conjugacy class). Each class corresponds to a physically distinct kind of symmetry (e.g., rotations about equivalent axes, or equivalent mirror planes).

Properties:

• The identity is always in a class by itself: X E X^{-1} = X X^{-1} = E.
• For abelian groups (AB = BA), n_{\text{classes}} = |G| — every element is its own class.

The three conjugacy classes of P(3): \{E\}, \{D, F\} (rotations), \{A, B, C\} (mirrors).

6) Cosets, Lagrange's Theorem, and Factor Groups

6.1) Cosets

For a subgroup H \subseteq G:

• Left coset: gH = \{gh \mid h \in H\} for some g \in G
• Right coset: Hg = \{hg \mid h \in H\} for some g \in G

A coset is not necessarily a subgroup. If g \in H, then the coset is just H itself (by the rearrangement theorem).

Two cosets of a given subgroup either have all the same elements or all different elements. This means the subgroup "divides up" (partitions) the group.

6.2) Lagrange's Theorem

The order of a subgroup is always a divisor of the order of the group.

where [G : H] is the number of distinct cosets (the index of H in G).

Proof sketch: Cosets of a subgroup partition the group — two cosets either have all the same elements or all different elements. Each coset has the same size as the subgroup (by the rearrangement theorem). So n \cdot |H| = |G| where n is the number of cosets.

For P(3) with |G| = 6: subgroups can only have order 1, 2, 3, or 6.

6.3) Homomorphisms, Normal Subgroups, and Factor Groups

A homomorphism is a many-to-one mapping that preserves multiplication. The kernel (elements mapping to the identity) must be a normal subgroup — one where left cosets equal right cosets (XNX^{-1} = N for all X).

When the kernel is normal, the cosets themselves form a factor group (quotient group) G/N, where each coset is treated as a single element. The order of the factor group is |G|/|N|.

Example: \{E, D, F\} is a normal subgroup of P(3). The factor group P(3)/\{E,D,F\} has two elements: \mathcal{E} = \{E, D, F\} and \mathcal{A} = \{A, B, C\}, forming P(2) \cong \mathbb{Z}_2.

Every group homomorphism corresponds to a factor group (first isomorphism theorem).

The interactive explorer below lets you visualize cosets and the connection between normal subgroups, homomorphisms, and factor groups.

7) Isomorphisms

Isomorphisms are "faithful" — they maintain a one-to-one correspondence between groups. Two groups are isomorphic if they have the same multiplication table (up to reordering of elements).

Example: We know the 3D permutation matrices and the 2D rotation+mirror matrices both represent P(3). We could have been given these matrices in any order — "isomorphic" means there exists a labeling that produces the same table.

Note that reordering the rows and columns of a multiplication table doesn't change the group — it's still the same table, just rearranged. An isomorphism goes further: it relabels elements, and the table structure (multiplication) must be preserved under this relabeling.

There are 6 valid isomorphisms from P(3) to itself (automorphisms), each corresponding to conjugation by a group element. Automorphisms always preserve conjugacy classes: rotations map to rotations, mirrors map to mirrors.

Related exercises: Groups Exercises