Groups: Definition, Examples, Theorems

Abstraction as Power

Nineteenth-century algebra made a central discovery: different mathematical objects (numbers, permutations, symmetries of polyhedra, matrices) obey the same abstract laws. By studying these laws in general form, we obtain results at once for all concrete cases.

Galois and Cauchy in the first half of the 19th century laid the foundations of group theory by studying symmetries of equations.

Definition of a Group

A group is a set $G$ with a binary operation $\ast$ satisfying:

1. Closure: $a \ast b \in G$ for all $a, b \in G$
2. Associativity: $(a \ast b) \ast c = a \ast (b \ast c)$
3. Identity element: there exists $e$ such that $a \ast e = e \ast a = a$
4. Inverse element: for each $a$ there exists $a^{-1}$ such that $a \ast a^{-1} = e$

If additionally $a \ast b = b \ast a$, the group is abelian (commutative).

Examples of Groups

($\mathbb{Z},+$) — the integers under addition: an infinite abelian group.

($\mathbb{R}^*,\cdot$) $=$ ($\mathbb{R}\setminus{0},\cdot$) — nonzero real numbers under multiplication.

($\mathbb{Z}/n\mathbb{Z},+$) $=$ ${0,1,\ldots,n-1}$ with addition modulo $n$ — a finite abelian group of order $n$.

The permutation group $S_n$ — all permutations of $n$ elements, operation is composition. $|S_n|=n!$. Not abelian for $n \geq 3$.

The symmetry group of the regular $n$-gon $D_n$ (dihedral group) of order $2n$: $n$ rotations and $n$ reflections.

$GL(n,\mathbb{R})$ — group of invertible real $n\times n$ matrices. $SL(n,\mathbb{R})$ — matrices with $\det = 1$.

Subgroups and Lagrange's Theorem

A subgroup $H \leq G$ is a subset that itself forms a group. $H \leq G \iff \emptyset \neq H \subseteq G$ and $\forall a,b \in H: a \ast b^{-1} \in H$.

Lagrange's theorem: If $G$ is a finite group and $H \leq G$, then $|H|$ divides $|G|$. The order of an element divides the order of the group.

Corollary: a group of prime order has no nontrivial subgroups — it is cyclic.

Homomorphism and Isomorphism

A homomorphism $\varphi: G \rightarrow H$ preserves the operation: $\varphi(a \ast b) = \varphi(a) \circ \varphi(b)$. Isomorphism is a bijective homomorphism. $G \cong H$ means “the same group with different notations”.

The kernel $\ker \varphi = {g: \varphi(g) = e_H}$ is a normal subgroup of $G$.