For a finite group G, a {\it normalizer covering} of G is a set of proper
normalizers of some subgroups of G whose union is G. First we give a
necessary and sufficient condition for a group having a {\it normalizer
covering}. Also, we find some properties of p-groups (p a prime) having a
normalizer covering. For a group G with a normalizer covering, we define
Οnβ(G) the minimum cardinality amongst all the normalizer coverings of
G. In this article, we show that if G is a p-group with a normalizer
covering, then Οnβ(G)=p+1 or 5. Finally, for any prime p and positive
integer k, we construct a solvable group G with Οnβ(G)=pk+1