# Notes - Groups HT23, Cyclic groups

### Flashcards

Can you define what it means for a group $G$ to be a cyclic group?

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

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?

*cyclic*group with order $n$. Breaking down $n$ into cases, what is $G$ isomorphic to?

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?

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.