Groups

Groups

A group $(G, *)$ is a set $G$ with a binary operation $*$ satisfying four axioms:

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

Example: $\mathbb{Z}_n$ under Addition

The set $\{0, 1, \ldots, n-1\}$ with addition modulo $n$ forms a group:

• Identity: $0$
• Inverse of $a$: $n - a$ (mod $n$)

def identity(n):
    return 0

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

Verifying Group Axioms

We can check whether a given Cayley table satisfies the group axioms computationally:

def has_identity(table):
    n = len(table)
    for e in range(n):
        if all(table[e][a] == a and table[a][e] == a for a in range(n)):
            return e
    return -1

Order of a Group

The order of a group is the number of elements: $|G|$. The group $\mathbb{Z}_n$ has order $n$.

Your Task

Implement verify_group(table) that takes a Cayley table (list of lists using elements $\{0, 1, \ldots, n-1\}$) and returns True if it defines a group. Check closure (all entries in range), associativity, identity, and inverses.