Every group of order $4$ is abelian: cyclic or elementary abelian.Suggest a slogan
Up to isomorphism, the groups of order \(4\) are \(\ZZ/4\ZZ\) and \((\ZZ/2\ZZ)^2\). Both are abelian.
Let \(\abs G=4\). If \(G\) has an element of order \(4\), then \(G\cong\ZZ/4\ZZ\). Otherwise every nonidentity element has order \(2\), so \(g^{-1}=g\) for all \(g\), and \(ab=(ab)^{-1}=b^{-1}a^{-1}=ba\) for all \(a,b\in G\). Then \(G\) is an abelian group of exponent \(2\), that is, written additively, a \(2\)-dimensional \(\FF_2\)-vector space, and \(G\cong(\ZZ/2\ZZ)^2\).
The cycle graphs of \(\ZZ/4\ZZ = \gens{g}\) and of \((\ZZ/2\ZZ)^2 = \theset{e, a, b, ab}\), with identity \(e\):
What are the groups of order 4?