Abstract: Vanishing theorems are playing a crucial role in the cohomology theory of groups in general for they determine cohomology groups upside a certain boundary. In this talk a specific vanishing theorem on the cohomology of an arithmetic group G with coefficients in the field of p elements (regarded as trivial G-module, where p is a prime number not dividing the order of any finite subgroup of G), using the virtual cohomological dimension of G (vcd G) as a boundary, will be presented. Useful background material including the transfer maps, cohomological dimension and some properties of arithmetic groups will be given. That the above theorem is not correct for any prime number p will be shown by a counterexample at the end of this speech.