Ways to Show a Group is Abelian

After some exposure to group theory, you quickly learn that when trying to prove a group $G$ is abelian, checking if $xy=yx$ for arbitrary $x,y$ in $G$ is not always the most efficient - or helpful! - tactic. Here is a (not comprehensive) running tab of other ways you may be able to prove your group is abelian:

  • Show the commutator $[x,y]=xyx^{-1}y^{-1}$of two arbitary elements $x,y \in G$ must be the identity
  • Show the group is isomorphic to a direct product of two abelian (sub)groups
  • Check if the group has order $p^2$ for any prime $p$ OR if the order is $pq$ for primes $p\leq q$ with $p\nmid q-1$.
  • Show the group is cyclic.
  • Show $|Z(G)|=|G|.$
  • Prove $G/Z(G)$ is cyclic. (e.g. does $G/Z(G)$ have prime order?)
  • Show that $G$ has a trivial commutator subgroup, i.e. is $[G,G]=\{e\}$.


Here's a thought map which is (probably) more fun than practical. Note, $p$ and $q$ denote primes below:

Reference: D. Dummit and R. Foote, Abstract Algebra, 3rd ed., Wiley, 2004. 

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