Daniel Herden, Montakarn Petapirak, José L. Rodríguez
August 17, 2017

### Abstract

A group homomorphism e:H→G{e:H\to G} is a cellular cover of G if for every homomorphism φ:H→G{\varphi:H\to G} there is a unique homomorphism φ¯:H→H{\bar{\varphi}:H\to H} such that φ¯e=φ{\bar{\varphi}e=\varphi}. Group localizations are defined dually. The main purpose of this paper is to establish 2ℵ0{2^{\aleph_{0}}} varieties of groups which are not closed under taking cellular covers. This will use the existence of a special Burnside group of exponent p for a sufficiently large prime p as a key witness. This answers a question raised by Göbel in [12]. Moreover, by using a similar witness argument, we can prove the existence of 2ℵ0{2^{\aleph_{0}}} varieties not closed under localizations. Finally, the existence of 2ℵ0{2^{\aleph_{0}}} varieties of groups neither closed under cellular covers nor under localizations is presented as well.