Let \(H\) be a Hopf algebra over the field \(k\). The authors show that an involutory Hopf algebra with non-zero integral verifying that there exists an injective indecomposable right comodule with dimension not divisible by the characteristic of \(k\) is cosemisimple. In particular, in characteristic zero, an involutory Hopf algebra with non-zero integral is always cosemisimple. This is a new proof of a result by \textit{J. B. Sullivan} [J. Algebra 19, 426-440 (1971; Zbl 0239.16006)]. Moreover, the following extension of a result of \textit{R. G. Larson} [J. Algebra 17, 352-368 (1971; Zbl 0217.33801)] is obtained: for a cosemisimple Hopf algebra, the characteristic of \(k\) does not divide the dimension of any absolutely irreducible right comodule.