Group

Intuition [Beginner]

A group is a system of reversible moves. You can do one move after another, there is a do-nothing move, and every move has an undo move.

Rotations of a square are a good first picture. Rotate by a quarter-turn, then rotate by a half-turn; the result is another rotation of the square. The do-nothing rotation leaves the square in place. Every rotation can be undone by rotating back.

Groups are the algebraic language of symmetry. Later units add geometry, topology, or smooth structure, but the reversible-move idea remains the base.

Visual [Beginner]

The four rotations of a square form a group. Combining two rotations gives another rotation.

The arrows record repeated quarter-turns. Four quarter-turns bring the square back to where it started.

Worked example [Beginner]

Use the clock positions $0,1,2,3$ to describe rotations of a square by quarter-turns. Add the numbers and wrap around after $3$.

For example, doing rotation $3$ and then rotation $2$ gives $1$, because five quarter-turns has the same final position as one quarter-turn.

The do-nothing move is $0$. The undo move for $1$ is $3$, because $1+3$ brings the square back to $0$.

What this tells us: a group is a closed system of reversible combinations.

Check your understanding [Beginner]

Formal definition [Intermediate+]

A group is a set $G$ with a binary operation

$$G\times G\to G,(g,h)\mapsto gh$$

such that:

1. $(gh)k=g(hk)$ for all $g,h,k\in G$.
2. There is $e\in G$ with $eg=ge=g$ for all $g\in G$.
3. For every $g\in G$ there is $g^{-1}\in G$ with $g{g}^{-1}=g^{-1}g=e$.

A group is abelian if $gh=hg$ for all $g,h\in G$. A homomorphism $\phi :G\to H$ is a function satisfying $\phi (gh)=\phi (g)\phi (h)$ ^{[Lang §I]}.

Key theorem with proof [Intermediate+]

Theorem (Kernel of a homomorphism is a subgroup). Let $\phi :G\to H$ be a group homomorphism. Then

$$\ker \phi =\{g\in G:\phi (g)=e_H\}$$

is a subgroup of $G$.

Proof. The identity lies in the kernel because $\phi (e_G)=e_H$. If $a,b\in \ker \phi$, then

$$\phi (a{b}^{-1})=\phi (a)\phi (b{)}^{-1}=e_H{e}_H^{-1}=e_H.$$

Thus $a{b}^{-1}\in \ker \phi$. The one-step subgroup criterion gives that $\ker \phi$ is a subgroup. $\square$

Bridge. The construction here builds toward 03.03.01 (lie group), where the same data is upgraded, and the symmetry side is taken up in 03.03.02 (group action). The defining pattern appears again in those units in a sharpened form, where the local data is glued or quotiented. Putting these together, the foundational insight is that the data of this unit gives the structural signature that the rest of the strand reads off.

Exercises [Intermediate+]

Lean formalization [Intermediate+]

lean_status: none is recorded for this curriculum unit because the project still needs stable local names connecting elementary group theory to group actions, Lie groups, and principal bundles. Mathlib itself has mature group-theory infrastructure.

Advanced results [Master]

The kernel construction is the first instance of a general principle: algebraic structure is measured by homomorphisms and their fibers over identity data. Normal subgroups are precisely the subgroups that can occur as kernels of homomorphisms, and quotient groups are the corresponding algebraic images ^{[Dummit-Foote §1]}.

Groups enter geometry through actions. A principal bundle is locally modeled on a group and has fibers on which the structure group acts freely and transitively 03.05.01. A Lie group is a group whose multiplication and inversion are smooth 03.03.01.

Synthesis. This construction generalises the pattern fixed in 00.02.05 (function), with the symmetric data replaced by its skew or twisted analogue. Read in the opposite direction, the construction is dual to the metric story: complements and orthogonality are taken with respect to the bilinear datum of this unit, not a metric, and the resulting category of subobjects is the one the rest of the strand classifies. The central insight is that this datum identifies algebra with geometry: functions become vector fields, subspaces become quotients, and invariants become cohomology classes — and that identification is the engine driving every theorem downstream.

Full proof set [Master]

If $N◃G$, the quotient set $G/N$ carries multiplication $(gN)(hN)=ghN$. This is well-defined because normality gives $h^{-1}Nh=N$, so changing representatives changes the product by an element of $N$. The projection $G\to G/N$ is a homomorphism with kernel $N$.

Conversely, if $N=\ker \phi$, then the computation in Exercise 7 proves $gN{g}^{-1}\subset N$ for every $g$. Applying the same inclusion to $g^{-1}$ gives equality, so $N$ is normal.

Connections [Master]

• Group actions 03.03.02 add a space on which a group acts. Orthogonal groups 03.03.03 are groups preserving a bilinear form. Lie groups 03.03.01 add smooth-manifold structure to a group, and principal bundles 03.05.01 use groups as structure groups acting on fibers.

• The spin group 03.09.03 and Virasoro algebra 03.11.03 are later symmetry objects whose definitions rely on this elementary algebraic foundation.

Historical & philosophical context [Master]

The modern group axioms arose from nineteenth-century work on permutations, equations, and geometry. Lang presents groups as the first algebraic structure because homomorphisms, kernels, and quotient constructions recur throughout algebra ^{[Lang §I]}.

Artin emphasizes groups as symmetry systems, while Dummit and Foote develop the subgroup and quotient machinery used throughout later algebra ^{[Artin Ch. 2; ref: TODO_REF Dummit-Foote §1]}.

Bibliography [Master]

• Serge Lang, Algebra, §I. ^{[Lang §I]}
• David Dummit and Richard Foote, Abstract Algebra, §1. ^{[Dummit-Foote §1]}
• Michael Artin, Algebra, Ch. 2. ^{[Artin Ch. 2]}