Generators

Overview

Let $G$ be a group and let $a_i\in G$ for $i\in I$ where $I$ is some index set. The smallest subgroup of $G$ containing $\{a_i\mid i\in I\}$ is the subgroup generated by $\{a_i\mid i\in I\}$.

If this subgroup is all of $G$, then $\{a_i\mid i\in I\}$ generates $G$. This is denoted $G=⟨a_i\mid i\in I⟩$. The set is called the generating set of $G$ and its members are generators of $G$.

If there is a finite set $\{a_i\mid i\in I\}$ that generates $G$, then $G$ is finitely generated.

Cyclic Groups

Let $G$ be a group and $a\in G$. Then the cyclic subgroup of $G$ generated by $a$ is defined as $$\langle a \rangle = {a^n \mid n \in \mathbb{Z}}.$$

This group is the minimum subgroup of $G$ that contains $a$. An element $a$ of a group $G$ generates $G$ if $⟨a⟩=G$. Alternatively, we call $a$ a generator for $G$. A group $G$ is cyclic if there is some element $a\in G$ that generates $G$.

Every cyclic group is abelian. Every subgroup of a cyclic group is cyclic. Every cyclic group of infinite order is isomorphic to $⟨\mathbb{Z},+⟩$. Every cyclic group of finite order $n\in {\mathbb{Z}}^{+}$ is isomorphic to $⟨{\mathbb{Z}}_n,{+}_n⟩$.

Let $G$ be a cyclic group with $n$ elements and generator $a$. Let $b\in G$ and let $b=a^s$. Then $b$ generates a cyclic subgroup $H$ of $G$ containing $n/d$ elements, where $d$ is the greatest common divisor of $n$ and $s$. Also, $⟨a^s⟩=⟨a^t⟩$ if and only if $gcd(s,n)=gcd(t,n)$.

Cyclic Isomorphisms

The group $\underset{i=1}{\overset{n}{⨁}}{\mathbb{Z}}_{m_i}$ is cyclic and isomorphic to ${\mathbb{Z}}_{m_1{m}_2\cdots m_n}$ if and only if $gcd(m_i,m_j)=1$ for all $1\le i<j\le n$. That is, $m_1,m_2,\dots ,m_n$ are pairwise coprime.

Fundamental Theorem of Finitely Generated Abelian Groups

Every finitely generated abelian group $G$ is isomorphic to a direct product of cyclic groups in the form

where the $p_i$ are primes, not necessarily distinct, and the $r_i$ are positive integers. The direct product is unique except for possible rearrangement of the factors. That is, the number of factors $\mathbb{Z}$, known as the Betti number of $G$, is unique and the prime powers $(p_i{)}^{r_i}$ are unique.

Cayley Digraphs

For each generating set $S$ of a finite group $G$, there exists a Cayley digraph. This is a digraph representing the group in terms of the generators in $S$. Every Cayley digraph must satisfy the following four properties:

1. It is connected.
2. At most one edge goes from any two vertices.
3. Every vertex $v$ has exactly one edge of each type incident from $v$ and exactly one edge incident to $v$.
4. If two different sequences of edge types start from $a$ and end at the same vertex $b$, the same sequences of edge types starting from any vertex $u$ will end at the same vertex $v$.

Conversely, any digraph satisfying these four properties is a Cayley digraph for some group.