GO Classes Test Series 2024 | Discrete Mathematics | Test 3 | Question: 15

Answer 1

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Lecture 7 - Lagrange's Theorem, Abelian Definitions & Practice | Group Theory | GO Classes - Discrete Mathematics

This is a very basic and standard result in group theory.

Let $\text{G}$ be a group and let $g \in \text{G}.$ The cyclic subgroup generated by $g$ is the subset :

$\langle g \rangle  = \{g^{n} \mid n \in \mathbb{Z}\}$

The following four statements are true for $\langle g \rangle:$

1. The cyclic subgroup $\langle g \rangle$ generated by $g$ is a subgroup of $\text{G}.$
2. $g \in \langle g \rangle.$
3. If $\text{H} \leq \text{G}$ and $g \in \text{H},$ then $\langle g \rangle \leq \text{H}.$ Hence $\langle g \rangle$ is the smallest subgroup of $\text{G}$ containing $g.$
4. $\langle g \rangle$ is abelian.