Groups and Subgroups - Applications

The subgroup lattice and fundamental counting theorems reveal deep structure in finite groups, with applications ranging from number theory to combinatorics.

TheoremSubgroup Criterion

Let $G$ be a group and $H \subseteq G$ be a non-empty subset. Then $H$ is a subgroup of $G$ if and only if for all $a, b \in H$ : $a \cdot b^{-1} \in H$

For finite groups, it suffices to check closure: if $H$ is finite and closed under the group operation, then $H$ is a subgroup.

This one-step criterion simplifies verification significantly. Instead of checking three conditions (identity, closure, inverses), we need only verify that $H$ is non-empty and closed under the operation $a \cdot b^{-1}$ .

TheoremIntersection of Subgroups

Let $\{H_i : i \in I\}$ be a collection of subgroups of $G$ . Then their intersection: $H = \bigcap_{i \in I} H_i$ is also a subgroup of $G$ .

The union of subgroups is generally NOT a subgroup.

ExampleGenerated Subgroups

For any subset $S \subseteq G$ , define the subgroup generated by $S$ as: $\langle S \rangle = \bigcap \{H \leq G : S \subseteq H\}$

This is the smallest subgroup containing $S$ . Explicitly: $\langle S \rangle = \{s_1^{\epsilon_1} s_2^{\epsilon_2} \cdots s_n^{\epsilon_n} : n \geq 0, s_i \in S, \epsilon_i \in \{-1, 1\}\}$

For a single element, $\langle g \rangle = \{g^n : n \in \mathbb{Z}\}$ is the cyclic subgroup generated by $g$ .

TheoremEuler's Theorem

If $\gcd(a, n) = 1$ , then: $a^{\phi(n)} \equiv 1 \pmod{n}$

where $\phi(n)$ is Euler's totient function counting integers coprime to $n$ less than $n$ .

This is a direct application of Lagrange's theorem to the multiplicative group $(\mathbb{Z}/n\mathbb{Z})^*$ of units modulo $n$ . Since this group has order $\phi(n)$ , every element $a$ satisfies $a^{\phi(n)} = 1$ .

Remark

Fermat's little theorem ( $a^{p-1} \equiv 1 \pmod{p}$ for prime $p$ ) is the special case where $\phi(p) = p - 1$ . These results underpin modern cryptography, particularly RSA encryption, which relies on the structure of $(\mathbb{Z}/n\mathbb{Z})^*$ for composite $n = pq$ .

The lattice of subgroups, ordered by inclusion, encodes the internal structure of $G$ . For cyclic groups, this lattice is particularly simple: it forms a chain corresponding to divisors of $|G|$ . Understanding subgroup lattices provides insight into quotient structures and solvability.