This page describes mathematical groups.

What is a group?

A group is a set G and a binary operation \cdot that together satisfy four properties: closure, associativity, identity, and inverse.

Closure: for any two elements a,b \in G, a \cdot b \in G.

Associativity: for any elements a,b,c \in G, (a \cdot b) \cdot c = a \cdot (b \cdot c).

Identity: there exists an element in G (often labelled 1 or e) called the identity element. For all elements a \in G, a \cdot e = a = e \cdot a.

Inverse: for every element a \in G, there exists some b \in G where a \cdot b = e.

Groups may be finite or infinite depending on whether the set G is finite or infinite.

Property: The identity element must be unique.

Suppose there exist two distinct identity elements e_1 and e_2 in a group G. For any element g, we know that g \cdot e_1 = g = e_1 \cdot g and g \cdot e_2 = g = e_2 \cdot g. So, e_2 \cdot e_1 = e_2 = e_1 \cdot e_2 and e_1 \cdot e_2 = e_1 = e_2 \cdot e_1. This means e_1 = e_2 \cdot e_1 = e_2, and thus e_1 = e_2.

Property: The inverse of an element is unique.

Let g be an element with two inverses a and b. We can show that a and b must be equivalent.

a = a \cdot e = a \cdot (g \cdot b) = (a \cdot g) \cdot b = e \cdot b = b

Examples of Groups

The integers under addition are an infinite group with 0 as the identity element.

All regular polygons are groups under fixed rotation that produces a symmetric shape. For instance, rotating a regular triangle by 120 degrees around the center results in a triangle symmetric to the original. This group is isomorphic to the cyclic group \mathbb{Z}_n.

Another similar example is the tetrahedron.

Subgroups and Homomorphisms

A subgroup is a subset H \subseteq G that still satisfies the group requirements under the same operation. (This implies that the identity element must be in H.)

The same language that's used for a "proper" subset applies to "proper" subgroups as well. If H does not include all elements in G, it is a proper subgroup, denoted H \subset G.

An example of a subgroup is the even integers under addition.

A homomorphism is a mapping f: G \rightarrow H to some other group H which preserves the group operation. More formally, for all elements a,b \in G, f(a \cdot b) = f(a) \cdot f(b).

Property: A group homomorphism must map the identity element in G to the identity element in H.

To see why, consider the contrary, that f(e_G) = h, where h \neq e_H. Then f(e_G \cdot e_G) = f(e_G) \cdot f(e_G) = h \cdot h = f(e_G). But since h is not e_H, h \cdot h \neq h.

Summarized from the wolfram pages on subgroups and group homomorphisms.

Special Types of Groups

Cyclic Group

The cyclic group \mathbb{Z}_n (or sometimes \mathbb{Z} / n \mathbb{Z}) represents rotations of fixed degree around a circle such that n rotations takes us back to the first position.

The cyclic group is isomorphic to the finite set of integers modulo n.

Cyclic groups can be entirely generated by a single element called the group generator.

Symmetric Group

TODO: This group is still a bit confusing. Let's try to tackle it bit by bit.

One definition: The symmetric group S_n has group elements which are all of the bijections from the set to itself. The group operator is function composition. The group order is n!. This definition comes from the wikipedia article on symmetric groups.

Wolfram gives an alternative definition: the symmetric group S_n of degree n is the group of all permutations on n symbols. What wasn't immediately clear (to me, at least) was that a bijection into the same set is a permutation. This is why the wiki definition gives as group elements bijective functions on a set of n elements.

Let's try to demonstrate this correspondence in a way that's a little more tactile with n=4. We can encode permutations as orderings of the digits 0, 1, 2, and 3. Exactly what this means is not immediately clear. We can, perhaps, say that the order of the digits defines the way in which the element "0123" is permuted to become the new encoding. So, "1023" means that the second element goes to the first position, the first element goes to the second position, and the third and fourth elements stay in place. This is clearly a bijection, since we're operating on unique elements, and each one is moved to a unique location.

Symmetry Group

TODO

TODOs

• Permutations on a set; the symmetric group.
• Symmetry groups as subgroups of general permutation groups.
• The Mobius group; cross-ratios, preservation of circles, the point at infinity
• Conjugation.
• Fixed points of Mobius maps and iteration.

Lagrange’s theorem

• Cosets.
• Lagrange’s theorem.
• Groups of small order (up to order 8).
• Quaternions.
• Fermat-Euler theorem from the group-theoretic point of view.

Group actions

• Group actions; orbits and stabilizers.
• Orbit-stabilizer theorem.
• Cayley’s theorem (every group is isomorphic to a subgroup of a permutation group).
• Conjugacy classes.
• Cauchy’s theorem.

Quotient groups

• Normal subgroups, quotient groups and the isomorphism theorem.

Matrix groups

• The general and special linear groups; relation with the Mobius group.
• The orthogonal and special orthogonal groups.
• Proof (in \mathbb{R}^3) that every element of the orthogonal group is the product of reflections and every rotation in \mathbb{R}^3 has an axis.
• Basis change as an example of conjugation.

Permutations

• Permutations, cycles and transpositions.
• The sign of a permutation.
• Conjugacy in S_n and in A_n.
• Simple groups; simplicity of A_5. [4]

Resources

M.A. Armstrong Groups and Symmetry. Springer–Verlag 1988

† Alan F Beardon Algebra and Geometry. CUP 2005

R.P. Burn Groups, a Path to Geometry. Cambridge University Press 1987

J.A. Green Sets and Groups: a first course in Algebra. Chapman and Hall/CRC 1988

W. Lederman Introduction to Group Theory. Longman 1976

Nathan Carter Visual Group Theory. Mathematical Association of America Textbooks