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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