# Tools of algebra (3): Groups

Algebra Level 5

How many of the following statements is/are true?

1. Any cyclic group is an abelian group.

2. Any simple abelian group is a cyclic group.

3. 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:

1. $$e$$ denotes the identity element in $$G$$
2. Abelian and ciclyc groups
3. Lagrange's Theorem
4. Normal subgroups
5. Simple groups
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