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is it enough…

that the presentation given by

\langle A,B\ :\ A^2=B^3,\quad A^2B=BA^2,\quad A^4=e,\quad B^6=e\rangle

determines the group SL_2({\mathbb{Z}}/{3\mathbb{Z}})?

Similar question for

\langle A,B,C:



A^4=B^6= C^2=e,


for the group GL_2({\mathbb{Z}}/{3\mathbb{Z}}).

Other similar problems but less “difficult” are:

  • \langle\varnothing:\varnothing\rangle=\{e\}
  • \langle A\ :\ A^2=e\rangle  for  \mathbb{Z}_2
  • \langle A\ :\ A^3=e\rangle  for  \mathbb{Z}_3
  • \langle A\ :\ A^4=e\rangle  for  \mathbb{Z}_4
  • \langle A,B\ :\ A^2=e, B^2=e, AB=BA\rangle  for  \mathbb{Z}_2\oplus\mathbb{Z}_2
  • \langle A,B\ :\ A^2=e, B^3=e, AB=B^2A\rangle  for  S_3
  • \langle A,B\ :\ A^2=e, B^3=e, AB=BA\rangle  for  \mathbb{Z}_2\oplus\mathbb{Z}_3
  • \langle A:\varnothing\rangle it is \mathbb{Z}
  • \langle A,B:AB=BA\rangle  is \mathbb{Z}\oplus\mathbb{Z}
  • \langle A,B:\varnothing\rangle  is \mathbb{Z}*\mathbb{Z}, the rank two free group
  • \langle A,B,J:(ABA)^4=e,ABA=BAB\rangle for SL_2(\mathbb{Z})
  • \langle A,B,J:(ABA)^4=J^2=e,ABA=BAB, JAJA=JBJB=e \rangle for GL_2(\mathbb{Z})
  • \langle A,B:A^2=B^3=e \rangle for P\!S\!L_2(\mathbb{Z})=SL_2(\mathbb{Z})/{\mathbb{Z}_2}\cong{\mathbb{Z}}_2*{\mathbb{Z}}_3
  • dare altri venti esempi
Qual è il tuo preferito?


Filed under algebra, free group, group theory, mathematics

a non orientable 3d-manifold with boundary

Mö x I

Mö x I

The picture on the right is representation and a description of its parts of  a 3d chunk, it is the trivial I-bundle over the möbius-strip, M\ddot{o}\times I. It is  useful to determine which 3d spaces are non orientable.

A 3d-space is non orientable if it has a simple closed curve whose 3d regular neighborhood is homeomorphic to the model of the picture.

If we denote by C\ddot{o} the core of the möbius strip, you can deduce for the curve C\ddot{o}\times\{\frac{1}{2}\} (the black pearled one) that its tangent bundle can’t be embedded in the tangent bundle of \mathbb{R}^3 or in any other tangent bundle of an orientable one.

Remember N_2 is the Klein bottle. Let’s the image talks by itself.

Ah, il nome di questo grande pezzo è la bottiglia solida di Klein.


Filed under fiber bundle, geometry, topology









L’enigma di Rubik e la sua soluzione.


Un gioco matematico, molto buffo ma realmente facile…

Che altro lei vuole che devo il discorso?

Ci sono qualcosa che dovrei scrivere?

 In che la lingua?  l’italiano è bello

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Filed under math