S. D. V. Stephens

notes / Studies in Algebra and Group Theory / chapter 1: Sets and Groups

1 Groups, Subgroups and Homomorphisms

· Denis Auroux

Original Professor Auroux’s handwritten notes (PDF), behind a password for now:

Course content

The course covers: (1) group theory (roughly Artin, chapter 2); (2) fields and vector spaces, linear and multilinear algebra (Axler); (3) more group theory (Artin, chapters 6–7); (4) an introduction to representation theory (Artin and Fulton–Harris). You should have Artin, Algebra (2nd edition), and Axler, Linear Algebra Done Right.

Groups

Groups are an abstract structure that models the common features of concrete objects such as

  • numbers,
  • permutations,
  • linear transformations,
  • symmetries.

Definition 1.1. A group consists of a set together with a law of composition, i.e. a map satisfying the following axioms:

  1. there exists an identity element such that , . ( means “for all”.) [Note: is unique! If both act as identity, then .]
  2. inverses exist: , such that . Write . ( means “there exists”.)
  3. associativity: , . [So we can write just .]

Remark.

  • Associativity (together with inverses) implies the cancellation law: , . (Proof: by associativity and the inverse axiom, so .)
  • Technically the group is the pair , but in real life we’ll just write for the set and talk of elements of .

Definition 1.2 (Variants).

  • If we omit the second axiom (inverses), we have a monoid. If we also omit the first (identity) and only keep associativity, we have a semigroup.
  • If we have a group whose law is commutative, i.e. for all , we say that is abelian (and may denote the operation instead).

Examples of groups

Example 1.3 (The trivial group). , with .

(Usually not an interesting example. Don’t give this as the answer to a HW problem asking for an example.)

Example 1.4 (Number systems). (the integers), or , , (rationals, reals, complex numbers) with addition. Identity: . Inverse: .

But the natural numbers only form a monoid!

Example 1.5 (A group with two elements?). If , let be the identity and the other element. Necessarily , , . What about ?

We can think of

  • , or , with addition mod 2 ();
  • with multiplication.

Question 1.6. Come up with an example of a group with 8 elements. Convince yourself it is a group. Can you find another example?

Example 1.7 (Integers mod ). with group law given by addition mod : (denote this by ). This is a finite group with elements. (We’ll see a better definition soon.)

Similarly, : with addition

Example 1.8 (Nonzero numbers under multiplication). , , with multiplication. Identity: . Inverse: .

Inside , the unit circle is also a group for multiplication.

These are still abelian. (Aside: the nonzero quaternions form a nonabelian multiplicative group.)

Example 1.9 (Symmetries and permutations). Recall that is

  • injective (one-to-one) if , ;
  • surjective (onto) if , such that ;
  • bijective if it is injective and surjective.

A permutation of a set is a bijection . The set of permutations of , with operation composition, is a group, . (Why?)

The symmetric group on elements is .

has a geometric interpretation if we think of the symmetries of an equilateral triangle: the rotations which preserve it (3, including the identity) and the reflections (3 of those).

An equilateral triangle with its three dashed reflection axes, each through a vertex and the midpoint of the opposite side, a rotation arrow around the centre, and small double arrows at the vertices indicating the reflections.

Symmetries permute the vertices, and every permutation of the set of vertices arises from exactly one symmetry (and the composition laws agree). So also occurs as the group of symmetries of .

(Other groups arise from symmetries of other geometric figures in and .)

Example 1.10 (Groups of matrices). with matrix multiplication: the “general linear group”. Also the “special linear group”. Also , for matrices with complex coefficients… or with or coefficients!

Notes not reached in this lecture, covered at the start of Lecture 2

Products of groups

Definition 1.11. Given two groups , the product group is with composition law .

  • If are finite, of order and , then is a finite group of order .
  • Similarly for a product of groups.

Example 1.12. , with (Similarly , , with componentwise addition.)

Given infinitely many groups , there are two different notions:

  • the direct product ;
  • the direct sum .

Example 1.13. Consider , and denote by . Then

Subgroups and homomorphisms

Definition 1.14. A subgroup of a group is a non-empty(!) subset which is closed under composition () and inversion (). Since , these 2 conditions imply . So (with the same operation) is a group in its own right.

We say is a proper subgroup if .

Definition 1.15. Given two groups , a homomorphism is a map which respects the composition law: , .

(This implies and .)

An isomorphism is a bijective homomorphism. (If and are isomorphic, then they are secretly the “same” group, even if the elements and the law may have different names.)

Question 1.16. Among the examples seen so far, which groups are isomorphic to each other? Or to subgroups of other groups?

Example 1.17 (Subgroups).

  • .
  • .
  • , the trivial subgroup.
  • , , and ??
  • .
  • .