Cyclic Group

Dependencies:

1. Subgroup
2. Conditions for a subset to be a subgroup

Let $G$ be a group and $g \in G$. Then the cyclic subgroup with generator $g$, denoted as $\langle g \rangle$, is $\{g^k: k \in \mathbb{Z}\}$.

A group $G$ is said to be cyclic if $\exists g \in G, \langle g \rangle = G$.

Proof that $\langle g \rangle$ is a subgroup of $G$

• Closure: $g^i * g^j = g^{i+j} \in \langle g \rangle$.
• Identity of $G$ is $e = g^0 \in \langle g \rangle$.
• Inverse of $g^k$ is $g^{-k}$, because $g^k * g^{-k} = g^{k + (-k)} = g^0 = e$.

Dependency for:

1. Subgroups of a cyclic group
2. Order of cyclic subgroup is order of generator
3. Group of prime order is cyclic
4. Order of elements in cyclic group
5. A cyclic group of order n has ϕ(n) generators
6. Cyclic groups are isomorphic to Z or Zn

Info:

• Depth: 3
• Number of transitive dependencies: 5

Transitive dependencies:

1. Group
2. Identity of a group is unique
3. Subgroup
4. Inverse of a group element is unique
5. Conditions for a subset to be a subgroup