Cyclic Groups Review

# Cyclic Groups Review

We will now review some of the recent material regarding cyclic groups.

• On the Generated Subgroups page we defined a new type of subgroup. Given a group $(G, *)$ and elements $a_1, a_2, ..., a_n \in G$, the Generated Cyclic Subgroup of $a_1, a_2, ..., a_n$ denoted $\langle a_1, a_2, ..., a_n \rangle$ is defined as the set of all elements in $G$ obtained from all possible products of the elements $a_1, a_2, ..., a_n$ and $a_1^{-1}, a_2^{-1}, ..., a_n^{-1}$.
• We noted that $\langle a_1, a_2, ..., a_n \rangle \subseteq G$ and since $\langle a_1, a_2, ..., a_n \rangle$ is closed under $*$ by definition, and the identity is contained in $\langle a_1, a_2, ..., a_n \rangle$ since $a_ka_k^{-1} \in \langle a_1, a_2, ..., a_n \rangle$, we have that $\langle a_1, a_2, ..., a_n \rangle$ is indeed a subgroup of $(G, *)$.
• On the Cyclic Groups page we said that a group $(G, *)$ is a Cyclic Group if there exists an element $a \in G$ such that:
(1)
\begin{align} \quad G = \langle a \rangle = \{ a^n : n \in \mathbb{Z} \} \end{align}
• In other words, a group $G$ is cyclic if there exists an element $a \in G$ such that every element in $G$ can be obtained as a power of $a$.
• We then looked at a very important result and a significant corollary. These are summarized below:
Property
(a) If $(G, *)$ is a cyclic group then $(G, *)$ is an abelian group.
(b) If $(G, *)$ is not an abelian group then $(G, *)$ is not a cyclic group.