Lovely little lemmas

Or maybe ‘amusing observations’

Rings are abelian

Posted on 14 November 2018 by Remy

In this post, we prove the following well-known lemma:

Lemma. Let $(R,+,\times,0,1)$ satisfy all axioms of a ring, except possibly the commutativity $a + b = b + a$ . Then $(R,+)$ is abelian.

That is, additive commutativity of a ring is implied by the other axioms.

Proof. By distributivity, we have $2(a+b) = 2a + 2b$ , so multiplication by $2$ is a homomorphism. By our previous post, this implies $R$ is abelian. $\qedsymbol$

Criteria for groups to be abelian

Posted on 14 November 2018 by Remy

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 $gh = hg$ . $\qedsymbol$