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?
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.