Cayley Table Generator
Generate operation tables for finite groups ($Z_n, U_n$) and visualize their structure.
Group Settings
Max recommended: 30 for readability.
Operation Table
Identity
Cayley Graph
Cyclic representation (Generator: 1)
Understanding Group Theory
A Group is a set of elements combined with an operation (like addition or multiplication) that satisfies four fundamental axioms:
- Closure: If you combine any two elements $a, b$ in the group, the result $a * b$ is also in the group.
- Associativity: The order of grouping doesn’t matter: $(a * b) * c = a * (b * c)$.
- Identity Element ($e$): There exists an element $e$ such that $a * e = a$ and $e * a = a$ for every element $a$. (Highlighted in Green above).
- Inverse Element: For every $a$, there is an element $a^{-1}$ such that $a * a^{-1} = e$.
Types of Groups in this Tool
$Z_n$ (Integers Modulo n): The set $\{0, 1, …, n-1\}$ under addition modulo $n$. This is always a cyclic group with identity 0.
$U(n)$ (Multiplicative Group): The set of integers less than $n$ that are coprime to $n$ (gcd is 1), under multiplication modulo $n$. The identity is 1. This group is essential in cryptography (e.g., RSA).
FAQ
What is a Cayley Table?
A Cayley Table is a grid that shows the result of the group operation for every pair of elements. It’s like a multiplication table but for abstract algebra. You can spot patterns like commutativity (symmetry across the diagonal) easily.
Why are some numbers missing in U(n)?
In $U(n)$, we only include numbers coprime to $n$. For example, in $U(6)$, the elements are $\{1, 5\}$ because 0, 2, 3, and 4 share factors with 6. If we included them, they wouldn’t have multiplicative inverses (e.g., $2 \times x \equiv 1 \pmod 6$ has no solution).
What is a Generator?
A generator is an element $g$ such that repeatedly applying the operation to $g$ eventually produces every element in the group. Groups with at least one generator are called Cyclic Groups.