In this paper, we introduce the notions of p-topological group and p-irresolute topological group which are generalizations of the notion topological group. We discuss the properties of p-topological groups with illustrative examples and we provide a connected p-topological group on any group G whose cardinality is not equal to 2. Also, we prove that translations and inversion in p-topological group are p-homeomorphism and demonstrate that every p-topological group is p-homogenous which leads to check whether a topology on a group satisfies the conditions of p-topological group or not.