Computing polycyclic quotients of finitely ( L -)presented groups via - PowerPoint PPT Presentation

Computing polycyclic quotients of finitely ( L -)presented groups via Gr¨ obner bases Max Horn joint work with Bettina Eick Technische Universit¨ at Braunschweig ICMS 2010, Kobe, Japan

Overview Polycyclic quotients of L -presented groups Quotient algorithms 1 Max Horn Quotient algorithms L -presented groups 2 L -presented groups Polycyclic Polycyclic quotient algorithm 3 quotient algorithm Gr¨ obner bases in group rings Gr¨ obner bases in group rings 4 Two examples Two examples 5

Overview Polycyclic quotients of L -presented groups Quotient algorithms 1 Max Horn Quotient algorithms L -presented groups 2 L -presented groups Polycyclic Polycyclic quotient algorithm 3 quotient algorithm Gr¨ obner bases in group rings Gr¨ obner bases in group rings 4 Two examples Two examples 5

Quotient Algorithms Polycyclic quotients of L -presented groups A quotient algorithm takes a group G (e.g. given via a Max Horn finite presentation) and computes a quotient H . Quotient algorithms An effective quotient map π : G → H is also computed, L -presented groups i.e., allowing computation of images and preimages. Polycyclic quotient algorithm H is ideally more tractable than G (e.g. finite or Gr¨ obner bases nilpotent), yet should share interesting features of G . in group rings Two examples Development and implementation of quotients methods for finitely presented groups have a long history.

Quotient Algorithms Polycyclic quotients of L -presented groups A quotient algorithm takes a group G (e.g. given via a Max Horn finite presentation) and computes a quotient H . Quotient algorithms An effective quotient map π : G → H is also computed, L -presented groups i.e., allowing computation of images and preimages. Polycyclic quotient algorithm H is ideally more tractable than G (e.g. finite or Gr¨ obner bases nilpotent), yet should share interesting features of G . in group rings Two examples Development and implementation of quotients methods for finitely presented groups have a long history.

Quotient Algorithms Polycyclic quotients of L -presented groups A quotient algorithm takes a group G (e.g. given via a Max Horn finite presentation) and computes a quotient H . Quotient algorithms An effective quotient map π : G → H is also computed, L -presented groups i.e., allowing computation of images and preimages. Polycyclic quotient algorithm H is ideally more tractable than G (e.g. finite or Gr¨ obner bases nilpotent), yet should share interesting features of G . in group rings Two examples Development and implementation of quotients methods for finitely presented groups have a long history.

Quotient Algorithms Polycyclic quotients of L -presented groups A quotient algorithm takes a group G (e.g. given via a Max Horn finite presentation) and computes a quotient H . Quotient algorithms An effective quotient map π : G → H is also computed, L -presented groups i.e., allowing computation of images and preimages. Polycyclic quotient algorithm H is ideally more tractable than G (e.g. finite or Gr¨ obner bases nilpotent), yet should share interesting features of G . in group rings Two examples Development and implementation of quotients methods for finitely presented groups have a long history.

Types of Quotient Algorithms Polycyclic quotients of Let G be a finitely presented group. Various quotient L -presented groups algorithms exist for such groups. They allow computing . . . Max Horn Quotient maximal abelian quotients, i.e., G / G ′ algorithms L -presented finite p -group quotients (Newman and O’Brien) groups Polycyclic finite solvable quotients (Niemeyer; Br¨ uckner and Plesken) quotient algorithm nilpotent quotients (Nickel) Gr¨ obner bases in group rings polycyclic quotients (Lo; most general in this sequence) Two examples ⇔ H is a polycyclic group ⇔ H is solvable and all subgroups are finitely generated ⇔ ∃ series H = H 1 � . . . � H n � 1 with H i / H i +1 cyclic

Types of Quotient Algorithms Polycyclic quotients of Let G be a finitely presented group. Various quotient L -presented groups algorithms exist for such groups. They allow computing . . . Max Horn Quotient maximal abelian quotients, i.e., G / G ′ algorithms L -presented finite p -group quotients (Newman and O’Brien) groups Polycyclic finite solvable quotients (Niemeyer; Br¨ uckner and Plesken) quotient algorithm nilpotent quotients (Nickel) Gr¨ obner bases in group rings polycyclic quotients (Lo; most general in this sequence) Two examples ⇔ H is a polycyclic group ⇔ H is solvable and all subgroups are finitely generated ⇔ ∃ series H = H 1 � . . . � H n � 1 with H i / H i +1 cyclic

Types of Quotient Algorithms Polycyclic quotients of Let G be a finitely presented group. Various quotient L -presented groups algorithms exist for such groups. They allow computing . . . Max Horn Quotient maximal abelian quotients, i.e., G / G ′ algorithms L -presented finite p -group quotients (Newman and O’Brien) groups Polycyclic finite solvable quotients (Niemeyer; Br¨ uckner and Plesken) quotient algorithm nilpotent quotients (Nickel) Gr¨ obner bases in group rings polycyclic quotients (Lo; most general in this sequence) Two examples ⇔ H is a polycyclic group ⇔ H is solvable and all subgroups are finitely generated ⇔ ∃ series H = H 1 � . . . � H n � 1 with H i / H i +1 cyclic

Types of Quotient Algorithms Polycyclic quotients of Let G be a finitely presented group. Various quotient L -presented groups algorithms exist for such groups. They allow computing . . . Max Horn Quotient maximal abelian quotients, i.e., G / G ′ algorithms L -presented finite p -group quotients (Newman and O’Brien) groups Polycyclic finite solvable quotients (Niemeyer; Br¨ uckner and Plesken) quotient algorithm nilpotent quotients (Nickel) Gr¨ obner bases in group rings polycyclic quotients (Lo; most general in this sequence) Two examples ⇔ H is a polycyclic group ⇔ H is solvable and all subgroups are finitely generated ⇔ ∃ series H = H 1 � . . . � H n � 1 with H i / H i +1 cyclic

Types of Quotient Algorithms Polycyclic quotients of Let G be a finitely presented group. Various quotient L -presented groups algorithms exist for such groups. They allow computing . . . Max Horn Quotient maximal abelian quotients, i.e., G / G ′ algorithms L -presented finite p -group quotients (Newman and O’Brien) groups Polycyclic finite solvable quotients (Niemeyer; Br¨ uckner and Plesken) quotient algorithm nilpotent quotients (Nickel) Gr¨ obner bases in group rings polycyclic quotients (Lo; most general in this sequence) Two examples ⇔ H is a polycyclic group ⇔ H is solvable and all subgroups are finitely generated ⇔ ∃ series H = H 1 � . . . � H n � 1 with H i / H i +1 cyclic

Types of Quotient Algorithms Polycyclic quotients of Let G be a finitely presented group. Various quotient L -presented groups algorithms exist for such groups. They allow computing . . . Max Horn Quotient maximal abelian quotients, i.e., G / G ′ algorithms L -presented finite p -group quotients (Newman and O’Brien) groups Polycyclic finite solvable quotients (Niemeyer; Br¨ uckner and Plesken) quotient algorithm nilpotent quotients (Nickel) Gr¨ obner bases in group rings polycyclic quotients (Lo; most general in this sequence) Two examples ⇔ H is a polycyclic group ⇔ H is solvable and all subgroups are finitely generated ⇔ ∃ series H = H 1 � . . . � H n � 1 with H i / H i +1 cyclic

Types of Quotient Algorithms Polycyclic quotients of Let G be a finitely presented group. Various quotient L -presented groups algorithms exist for such groups. They allow computing . . . Max Horn Quotient maximal abelian quotients, i.e., G / G ′ algorithms L -presented finite p -group quotients (Newman and O’Brien) groups Polycyclic finite solvable quotients (Niemeyer; Br¨ uckner and Plesken) quotient algorithm nilpotent quotients (Nickel) Gr¨ obner bases in group rings polycyclic quotients (Lo; most general in this sequence) Two examples ⇔ H is a polycyclic group ⇔ H is solvable and all subgroups are finitely generated ⇔ ∃ series H = H 1 � . . . � H n � 1 with H i / H i +1 cyclic

Types of Quotient Algorithms Polycyclic quotients of Let G be a finitely presented group. Various quotient L -presented groups algorithms exist for such groups. They allow computing . . . Max Horn Quotient maximal abelian quotients, i.e., G / G ′ algorithms L -presented finite p -group quotients (Newman and O’Brien) groups Polycyclic finite solvable quotients (Niemeyer; Br¨ uckner and Plesken) quotient algorithm nilpotent quotients (Nickel) Gr¨ obner bases in group rings polycyclic quotients (Lo; most general in this sequence) Two examples ⇔ H is a polycyclic group ⇔ H is solvable and all subgroups are finitely generated ⇔ ∃ series H = H 1 � . . . � H n � 1 with H i / H i +1 cyclic