Advanced Group Theory

Advanced group theory relies heavily on understanding how subgroups partition a group and how these partitions can form new mathematical structures.

Normal Subgroups

When working with a subgroup $N$ inside a larger group $G$, we can form subsets called cosets. For any element $g$ in $G$, the left coset is the set formed by multiplying $g$ with every element in $N$ ($gN$). The right coset is formed by multiplying in the opposite order ($Ng$).

A normal subgroup is a special type of subgroup where it is invariant under conjugation. Formally, this means that $gng^{-1}$ is always an element of $N$ for any $n$ in $N$ and any $g$ in $G$. An equivalent and highly useful way to define a normal subgroup is that its left and right cosets are identical: $gN = Ng$ for every element $g$ in the group.

def is_normal(G, H, op):
  for g in G:
      left_coset = set([op(g, h) for h in H])
      right_coset = set([op(h, g) for h in H])
      if left_coset != right_coset:
          return False
  return True

# In Abelian groups, all subgroups are normal
G = list(range(12))
H = [0, 4, 8]
add_mod_12 = lambda a, b: (a + b) % 12
print(f"Is {H} normal in Z/12Z? {is_normal(G, H, add_mod_12)}")

Quotient Groups

The primary reason normal subgroups are so important is that they allow the construction of quotient groups (or factor groups).

If $N$ is a normal subgroup of $G$, the quotient group, denoted as $G/N$, is defined as the set of all left cosets of $N$ inside $G$. The elements of this new group are entire sets (the cosets). The group operation is defined by multiplying the representatives of the cosets: $(aN)(bN) = (ab)N$. This operation is only mathematically valid and well-defined when $N$ is normal.

Lagrange’s Theorem

Lagrange’s Theorem describes a fundamental limitation on the size of subgroups within finite groups. The theorem states that if a group $G$ has a finite number of elements (its order, denoted $|G|$), then the order of any subgroup $H$ must perfectly divide the order of $G$.

The number of distinct cosets of $H$ in $G$ is called the index of $H$, written as $[G : H]$. Lagrange’s theorem proves that the total size of the group is the product of the subgroup’s size and its index: $|G| = [G : H] \cdot |H|$

def get_cosets(G, H, op):
  cosets = set()
  for g in G:
      # Left coset gH = {g * h for h in H}
      coset = frozenset([op(g, h) for h in H])
      cosets.add(coset)
  return cosets

cosets = get_cosets(G, H, add_mod_12)
print(f"Number of cosets (index): {len(cosets)}")
print(f"Subgroup size: {len(H)}")
print(f"Group size: {len(G)}")
print(f"Does |H| divide |G|? {len(G) % len(H) == 0}")

References & Further Reading

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