Groups and homomorphisms

Editorially prepared introduction for this redesigned journal. This is new explanatory content, separate from Pablo's original notes.

1. Groups

Groups abstract the symmetries and reversible operations found throughout mathematics.

Definition. A group is a set \(G\) with a binary operation \(\cdot\) satisfying associativity, possessing an identity \(e\), and providing an inverse \(g^{-1}\) for every \(g\in G\).

Example. \((\mathbb Z,+)\) is a group with identity \(0\); the inverse of \(n\) is \(-n\). The nonzero real numbers form a group under multiplication.

2. Subgroups

A subset \(H\subseteq G\) is a subgroup if it forms a group under the inherited operation.

Subgroup test. A nonempty subset \(H\) is a subgroup if \(ab^{-1}\in H\) for all \(a,b\in H\).

3. Homomorphisms

A group homomorphism \(\varphi:G\to K\) respects multiplication: \(\varphi(ab)=\varphi(a)\varphi(b)\).

Its kernel \(\ker\varphi=\{g\in G:\varphi(g)=e_K\}\) is a normal subgroup of \(G\); its image is a subgroup of \(K\).

4. A useful theorem

First isomorphism theorem. Every homomorphism \(\varphi:G\to K\) induces an isomorphism \(G/\ker\varphi\cong\operatorname{im}\varphi\).

Example. The map \(\mathbb Z\to\mathbb Z/n\mathbb Z\) taking an integer to its residue class has kernel \(n\mathbb Z\).

Groups and homomorphisms.

1. Groups

2. Subgroups

3. Homomorphisms

4. A useful theorem