Samjhake
Group Theory

Group Theory· L01

Groups & subgroups

Axioms, examples, cyclic groups.

Group

A set GG with an associative binary operation, an identity, and inverses.

P1

Subgroups of cyclic groups

Show that every subgroup of a cyclic group is cyclic.

Solution · P1
  1. If G=⟨a⟩G=\langle a\rangle and H≤GH\le G, take the least positive kk with ak∈Ha^k\in H.
  2. Prove H=⟨ak⟩H=\langle a^k\rangle.