# Tools of algebra (3): Groups

**Algebra**Level 5

How many of the following statements is/are true?

Any cyclic group is an abelian group.

Any simple abelian group is a cyclic group.

Let \(G\) be a finite group, then the following are equivalent:

a) \(|G| \) is prime.

b) \(G\) and \(\{e\}\) are the only subgroups in \(G\), and \(G \neq \{e\}\).

c) \(G\) is a cyclic group and \(G \cong \mathbb{Z}_p\) for some \(p\) prime.

Relevant notes:

- \(e\) denotes the identity element in \(G\)
- Abelian and ciclyc groups
- Lagrange's Theorem
- Normal subgroups
- Simple groups