ThmDex – An index of mathematical definitions, results, and conjectures.
Result R4279 on D1301: Generated subgroup
Explicit algebraic expression for elements of generated subgroup with singleton generator set
Formulation 0
Let $G$ be a D22: Group such that
(i) $E \subseteq G$ is a D78: Subset of $G$
(ii) $E = \{ g \}$ is a D135: Singleton set
(iii) $\langle E \rangle$ is a D1301: Generated subgroup of $G$ with generator $E$
Then \begin{equation} \langle E \rangle = \{ g^n : n \in \mathbb{Z} \} \end{equation}
Proofs
Proof 0
Let $G$ be a D22: Group such that
(i) $E \subseteq G$ is a D78: Subset of $G$
(ii) $E = \{ g \}$ is a D135: Singleton set
(iii) $\langle E \rangle$ is a D1301: Generated subgroup of $G$ with generator $E$
This result is a particular case of R1450: Explicit algebraic expression for elements of generated subgroup. $\square$