Criteria for groups to be abelian

This is a review of some elementary criteria for a group to be abelian.

Lemma. Let $G$ be a group. Then the following are equivalent:

$G$ is abelian,

The map $G \to G$ given by $g \mapsto g^2$ is a group homomorphism;

The map $G \to G$ given by $g \mapsto g^{-1}$ is a group homomorphism;

The diagonal $G \subseteq G \times G$ is normal.

Proof. We prove that each criterion is equivalent to (1).

For (2), note that $(gh)^2 = ghgh$ , which equals $gghh$ if and only if $gh = hg$ .

For (3), note that $(gh)^{-1} = h^{-1}g^{-1}$ , which equals $g^{-1}h^{-1}$ if and only if $gh = hg$ .

For (4), clearly $\Delta_G \colon G \hookrightarrow G \times G$ is normal if $G$ is abelian. Conversely, note that $(e,h)(g,g)(e,h^{-1}) = (g,hgh^{-1})$ , which is in the diagonal if and only if $gh = hg$ . $\qedsymbol$

3 thoughts on “Criteria for groups to be abelian”

The map $G \to G$ given by $g \mapsto g^n$ is a group homomorphism for $n = 28,50,81$ .

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Remy on 1 September 2019 at 20:33 said:

Fascinating; I have never heard of this. Are you saying that this is true for any group $G$ ? Do you know where I can read more about this?

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- fvenez on 4 September 2019 at 19:12 said:
  
  Hello, Remy 🙂
  Yes, I meant that any group $G$ such that the three maps $g\mapsto g^{28}$ , $g\mapsto g^{50}$ , $g\mapsto g^{81}$ are three group endomorphisms must be an abelian group.
  
  You can find conditions of this type among the exercises in Herstein’s algebra textbook (which is now a bit too old-fashioned, but contains a fantastic wealth of good exercises). For example you can prove that if $g\mapsto g^{n}$ is an endomorphism for three consecutive values of $n$ , then $G$ must be abelian; or that if $g\mapsto g^{3}$ and $g\mapsto g^{5}$ are both endomorphisms, then $G$ must be abelian.
  I liked these exercises a lot as a student, so I was very pleased to see the topic addressed fully in this nice elementary paper:
  Abelian Forcing Sets, Joseph A. Gallian and Michael Reid, The American Mathematical Monthly, Vol. 100, No. 6 (Jun. – Jul., 1993), pp. 580-582
  
  Reply ↓

Lovely little lemmas

Or maybe ‘amusing observations’

3 thoughts on “Criteria for groups to be abelian”

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