Example and Counterexamples: Cyclic groups

Study of cyclic groups form a crucial part in group theory. Here are some important definitions, facts and examples of cyclic groups.

Cyclic group and its generator

Let G be a group and a \in G. Then the set <a>=\{a^n : n\in \Bbb Z\} forms a subgroup of G and is called the subgroup generated by a.

If for some a\in G, <a>=G, then G is called a cyclic group and a is referred to as it’s generator.

Examples

  1. (\Bbb Z,+)
  2. Integer modulo n, ( \Bbb Z /n \Bbb Z)
  3. \Bbb Z_m \times \Bbb Z_n whenever Gcd(m,n)=1
  4. A_n, for n\le 3

Counterexamples

  1. (\Bbb Q,+)
  2. (\Bbb R,+)
  3. \Bbb Z_p \times \Bbb Z_p , where p is any prime.
  4. \Bbb Q/ \Bbb Z
  5. Any non-abelian group.

Important facts

  1. Any cyclic group is abelian.
  2. Any cyclic group is either finite or countable.
  3. Any subgroup of cyclic group is cyclic.
  4. Any quotient group of cyclic group is cyclic.
  5. Any group of prime order is cyclic.
  6. Any group G, with o(G)=pq, p<q, p\nmid q-1, where p and q are distinct primes is cyclic.
  7. Any non-abelian group is non-cyclic.
  8. Any group which is uncountable is not cyclic.
  9. If a \in G is a generator of G, then a^-1 is also a generator of G.
  10. Any infinite cyclic group has exactly two generators and is isomorphic to (\Bbb Z,+).
  11. Any finite cyclic group G is isomorphic to \Bbb Z_n, where n=o(G).

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