Group Theory Fundamentals

Group theory is the mathematical study of symmetry. It provides a formal framework for analyzing operations that preserve the structure of an object. Formulated in the 19th century by Evariste Galois and Niels Henrik Abel, it has become the foundational language of modern algebra, theoretical physics (e.g., the Standard Model), and cryptography.

Formal Definition of a Group

A group $(G,\cdot )$ is a set $G$ together with a binary operation $\cdot :G\times G\to G$ that satisfies the following four axioms:

1. Closure: For all $a,b\in G$, the element $a\cdot b$ is in $G$.
2. Associativity: For all $a,b,c\in G$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
3. Identity Element: There exists an element $e\in G$ such that for every $a\in G$, $e\cdot a=a\cdot e=a$.
4. Inverse Element: For each $a\in G$, there exists an element $b\in G$ (denoted $a^{-1}$) such that $a\cdot b=b\cdot a=e$.

If the operation also satisfies $a\cdot b=b\cdot a$ for all $a,b\in G$, the group is called Abelian (or commutative).

Diversity of Groups: Examples

• Additive Group of Integers $(\mathbb{Z},+)$: Identity is $0$, inverse of $n$ is $-n$. This is a cyclic Abelian group.
• Multiplicative Group of Non-zero Rationals $({\mathbb{Q}}^{\ast },\times )$: Identity is $1$, inverse of $q$ is $1/q$.
• General Linear Group $G{L}_n(\mathbb{R})$: The group of $n\times n$ invertible matrices under matrix multiplication. This is a non-Abelian group for $n\ge 2$.
• Symmetric Group $S_n$: The group of all permutations of $n$ elements. $|S_n|=n!$.
• Dihedral Group $D_n$: The group of symmetries of a regular $n$-gon.

Subgroups

A subset $H$ of a group $G$ is called a subgroup ($H\le G$) if $H$ itself forms a group under the operation inherited from $G$.

The Subgroup Criterion

A non-empty subset $H\subseteq G$ is a subgroup if and only if for all $a,b\in H$, $a\cdot b^{-1}\in H$.

Cyclic Groups and Order

The order of a group $G$, denoted $|G|$, is its cardinality. The order of an element $g\in G$ is the smallest positive integer $n$ such that $g^n=e$. If no such $n$ exists, $g$ has infinite order.

Cyclic Groups

A group $G$ is cyclic if there exists an element $g\in G$ such that every $x\in G$ can be written as $g^k$ for some integer $k$. We write $G=⟨g⟩$.

• Finite example: $\mathbb{Z}/n\mathbb{Z}$ under addition.
• Infinite example: $\mathbb{Z}$ under addition (generated by $1$ and $-1$).

Group Homomorphisms

A homomorphism is a mapping between groups that preserves the group operation. Let $(G,\cdot )$ and $(H,\ast )$ be groups. A function $f:G\to H$ is a homomorphism if for all $a,b\in G$: $f(a\cdot b)=f(a)\ast f(b)$

Kernel and Image

• Kernel: $\ker (f)=\{g\in G\mid f(g)=e_H\}$. The kernel is always a normal subgroup of $G$.
• Image: $Im(f)=\{f(g)\mid g\in G\}$. The image is a subgroup of $H$.

An isomorphism is a bijective homomorphism. If an isomorphism exists between $G$ and $H$, we say $G\cong H$, meaning they are structurally identical as groups.

Cayley’s Theorem

Every group $G$ is isomorphic to a subgroup of the symmetric group acting on $G$. Specifically, every group can be viewed as a group of permutations. This theorem highlights the universality of permutation groups in the study of finite symmetry.

Computational Representation of Groups

In software, groups are often represented by their multiplication tables (for small finite groups) or by their generators and relations.

class IntegerGroupModN:
    def __init__(self, n):
        self.n = n
        self.elements = list(range(n))

    def op(self, a, b):
        return (a + b) % self.n

    def inverse(self, a):
        return (self.n - a) % self.n

    def is_subgroup(self, subset):
        # Simplified check for closure and inverses
        for a in subset:
            for b in subset:
                if self.op(a, self.inverse(b)) not in subset:
                    return False
        return True

# G = Z/6Z
G = IntegerGroupModN(6)
H = [0, 2, 4] # Subset {0, 2, 4}
print("Is %s a subgroup of Z/6Z? %s" % (H, G.is_subgroup(H)))

Exercise