Group Theory Fundamentals

In mathematics, a group is a foundational structure used to study symmetry and the behavior of operations.

Formal Definition of a Group

As defined by standard algebra, a group consists of an ordered pair $(G, \cdot)$. This pairs a set of elements, $G$, with a binary operation, typically denoted as "$\cdot$", which combines any two elements to produce a third.

For this pairing to qualify as a group, it must strictly satisfy three axioms:

1. Associativity: Grouping does not affect the outcome. For any elements $a, b,$ and $c$, the equation $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ must hold true.
2. Identity Element: The set must contain a unique element $e$, such that combining it with any element $a$ leaves $a$ unchanged: $e \cdot a = a$ and $a \cdot e = a$.
3. Inverse Element: Every element $a$ must have a corresponding inverse element, often written as $a^{-1}$. Combining an element with its inverse always yields the identity element: $a \cdot a^{-1} = e$ and $a^{-1} \cdot a = e$.

Additionally, the operation must be closed over the set, meaning the result of $a \cdot b$ is always an element within $G$. If the operation is also commutative ($a \cdot b = b \cdot a$), the group is known as an Abelian group.

def check_group_axioms(elements, op, identity):
  # 1. Closure (Simplified for finite set)
  for a in elements:
      for b in elements:
          if op(a, b) not in elements: return False, "Closure failed"
  
  # 2. Identity
  for a in elements:
      if op(a, identity) != a or op(identity, a) != a:
          return False, f"Identity failed for {a}"
  
  # 3. Inverses
  for a in elements:
      found_inverse = False
      for b in elements:
          if op(a, b) == identity and op(b, a) == identity:
              found_inverse = True
              break
      if not found_inverse: return False, f"Inverse failed for {a}"
      
  return True, "All axioms satisfied"

# Test Z/4Z under addition
z4 = [0, 1, 2, 3]
add_mod_4 = lambda a, b: (a + b) % 4
print(f"Z/4Z result: {check_group_axioms(z4, add_mod_4, 0)}")

# Test Z/4Z under multiplication (Identity = 1)
mult_mod_4 = lambda a, b: (a * b) % 4
print(f"Z/4Z mult result: {check_group_axioms(z4, mult_mod_4, 1)}")

Subgroups and Isomorphisms

A subset $H$ of a group $G$ that independently forms a group under the same operation is called a subgroup. To verify if $H$ is a subgroup, one can check if $g^{-1} \cdot h$ remains in $H$ for all elements $g, h$ in $H$.

When analyzing the relationship between two groups, we use a mapping called a homomorphism. This function, $\varphi: G \to H$, preserves the structural integrity of the operations such that $\varphi(a \cdot b) = \varphi(a) * \varphi(b)$. If this mapping is bijective (a perfect one-to-one correspondence), it is called an isomorphism, indicating the two groups are structurally identical.

References & Further Reading

• Wikipedia: Group (mathematics) (CC-BY-SA 3.0)