OPENAIRE OpenAire zbMATH Open Closed Access Let (p ) be a prime integer. Denote by (M_ n ) the free metabelian group of finite rank (n ) and by (M_{n,p} ) the group (M_ n/[M_ n,M_ n]^ p ). Let (M_ n(p) ) be the free metabelian pro- (p )-group of rank (n ) and (M_{n,p}(p)=M_ n(p)/ [M_ n(p),M_ n(p)]^ p ). Then (M_ n(p) ) is a (p )-adic completion of (M_ n ) and (M_{n,p}(p) ) is a (p )- adic completion of (M_ n(p) ). textit{V. N. Remeslennikov} [Izv. Akad. Nauk SSSR, Ser. Mat. 43, 399-417 (1979; Zbl 0414.20027)] found a Magnus- type representation of (M_ n(p) ) by (2 times 2 )-matrices. In this paper a similar representation for (M_{n,p}(p) ) by means of modules over an appropriate ring is given as well as a Bachmuth-type representation of IA-automorphisms of (M_{n,p}(p) ) and (M_ n(p) ). By an IA-automorphism is meant an automorphism which induces the identical automorphism modulo the commutator subgroup. One of the main results of the paper shows that for (n geq 2 ) the group of IA-automorphisms of (M_{n,p}(p) ) and of (M_ n(p) ) is not finitely generated. The group (G ) of automorphisms of (M_{n,p}(p) ), (n neq 3 ), is finitely generated. An explicit form of a finite generating set in (G ) is given. Generators, relations, and presentations of groups finite generating set Bachmuth-type representation Solvable groups, supersolvable groups Free nonabelian groups group of IA-automorphisms IA- automorphisms Automorphisms of infinite groups Magnus-type representation Automorphism groups of groups (p )-adic completion free metabelian pro- (p )-group Limits, profinite groups free metabelian group Roman'kov, V. A. 1992-01-01 Let (p ) be a prime integer. Denote by (M_ n ) the free metabelian group of finite rank (n ) and by (M_{n,p} ) the group (M_ n/[M_ n,M_ n]^ p ). Let (M_ n(p) ) be the free metabelian pro- (p )-group of rank (n ) and (M_{n,p}(p)=M_ n(p)/ [M_ n(p),M_ n(p)]^ p ). Then (M_ n(p) ) is a (p )-adic completion of (M_ n ) and (M_{n,p}(p) ) is a (p )- adic completion of (M_ n(p) ). textit{V. N. Remeslennikov} [Izv. Akad. Nauk SSSR, Ser. Mat. 43, 399-417 (1979; Zbl 0414.20027)] found a Magnus- type representation of (M_ n(p) ) by (2 times 2 )-matrices. In this paper a similar representation for (M_{n,p}(p) ) by means of modules over an appropriate ring is given as well as a Bachmuth-type representation of IA-automorphisms of (M_{n,p}(p) ) and (M_ n(p) ). By an IA-automorphism is meant an automorphism which induces the identical automorphism modulo the commutator subgroup. One of the main results of the paper shows that for (n geq 2 ) the group of IA-automorphisms of (M_{n,p}(p) ) and of (M_ n(p) ) is not finitely generated. The group (G ) of automorphisms of (M_{n,p}(p) ), (n neq 3 ), is finitely generated. An explicit form of a finite generating set in (G ) is given. Generating elements of automorphism groups of free metabelian pro-\(p\)- groups d72741b4015cc9416292ca697c6d1880 publication https://doi.org/d72741b4015cc9416292ca697c6d1880