Central Extension Of Degree P Of A₂-Group Generated By Three Elements And Its Applications

Let N and H be arbitary groups. If there exists a group G which has a normal subgroup (?) ≤ Z(G) such that (?)= N and G/(?) = H, then G is called a central extension of N by H. In particular, if|N|=p, then G is called a central extention of H of degree p. Suppose the Gbea finite p-group. If all subgroups of index pt of G are ablian and at least one subgroup of index pt-1 of G is not ablian, then G is called an,Argroup. In this paper, we give the central extension of degree p of A2-group generated by three elements. Moreover, we classify finite p-groups generated by three elements whose proper quotient groups are abelian or minimal non-abelian or A2-group.