# Notes - Groups HT23, Cyclic groups

> Source: https://ollybritton.com/notes/uni/prelims/ht23/groups/notes/cyclic-groups/ · Updated: 2023-02-17 · Tags: uni, notes

- [Course - Groups and Group Actions HT23](https://ollybritton.com/notes/uni/prelims/ht23/groups/)
	- [Notes - Groups HT23, Subgroups](https://ollybritton.com/notes/uni/prelims/ht23/groups/notes/subgroups/)

### Flashcards
Can you define what it means for a group $G$ to be a cyclic group?::
$$
\exists a \in G \text{ s.t. } G = \\{a^n : n \in \mathbb Z\\}
$$

What generator gives rise to the group $(\mathbb Z, +)$?::
$$
1
$$

How can you form a cyclic subgroup of any group $G$ from a single element $g$?::
Let
$$
H = \\{ g^n : n \in \mathbb Z \\} \subseteq G
$$

Let $G$ be a *cyclic* group with order $n$. Breaking down $n$ into cases, what is $G$ isomorphic to?::
$$
G \cong \begin{cases}
C_n&\text{if }n \text{ is finite} \\\\
(\mathbb Z, +) &\text{otherwise}
\end{cases}
$$

What is true about any subgroup of a cyclic group?::
It is also cyclic.

What’s a common proof technique used for showing a group is isomorphic to a cyclic group?::
Find a generator with the same order as the size of the group.

Let $G = \langle g \rangle$ and $H \leqslant G$. When proving that the subgroup of a cyclic group is also cyclic, what value $n$ do you define that you later go on to show is a generator?::
$$
n = \min \\{k > 0 : g^k \in H\\}
$$

Let $G = \langle g \rangle$ and $H \leqslant G$. When proving that any subgroup of a cyclic group is also cyclic, you show $H = \langle g^n \rangle$ where
$$
n = \min \\{k > 0 : g^k \in H\\}
$$
Suppose some $g^a \in H$. Can you quickly show that $g^a = (g^n)^q$ for some $q \in \mathbb Z$?::
$\exists q, r \in \mathbb Z$ such that $a = qn + r$ where $0 \le r < n$. Therefore
$$
g^r = g^{a - qn} = g^a (g^n)^{-q} \in H
$$
so $g^r \in H$. But since $n$ is minimal, $r$ has to be zero otherwise $n = r$ which is impossible. Hence
$$
g^a = (g^n)^q
$$

### Proofs
Prove that any subgroup of a cyclic group is also cyclic.::
Todo.

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