Varieties of groups and cellular covers
A group homomorphism e : H ---> G is a cellular cover of G if for any homomorphism φ in Hom(H,G) there is a unique homomorphism ψ : H ---> H such that ψe = φ. A. Yu. Ol’shanskii established in 1970 the existence of a continuum of varieties of groups. We modify his result and show that there exists a continuum of pairwise distinct varieties of groups which are not closed under taking cellular covers by applying the existence of a special Burnside group of exponent p for a sufficiently large prime p. This answers a question raised by R. Göbel [Forum Math., Vol. 24 (2012), 317-337]. Moreover, by similar arguments we can conclude a dual result for localizations which are defined similarly to cellular covers as group homomorphisms e : G ---> H such that for any homomorphism φ in Hom(G,H) there is a unique homomorphism ψ : H ---> H with eψ = φ.