In this paper we define the pullback cat1-group and show that this Pullback has a right adjoint which is the induced cat1-group. Later we show that this right adjoint is a pushout of category of cat1-groups. We calculate the Peiffer subgroups to find a finite group of the source of induced cat1-groups. The generating set of Peiffer subgroups are also given in this paper. All results are corrected by a GAP[13] program package in [4]. This paper also contains the some computational examples which are the calculation-induced cat1-group and comparative times between the induced crossed modules and induced cat1-groups.

In this paper we define the pullback cat1-group and show that this Pullback has a right adjoint which is the induced cat1-group. Later we show that this right adjoint is a pushout of category of cat1-groups. We calculate the Peiffer subgroups to find a finite group of the source of induced cat1-groups. The generating set of Peiffer subgroups are also given in this paper. All results are corrected by a GAP[13] program package in [4]. This paper also contains the some computational examples which are the calculation-induced cat1-group and comparative times between the induced crossed modules and induced cat1-groups.