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replace Z with Q because c is not in GL_9(Z), add "elements" because edits must be at least 6 characters - wtf?!
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Let $G$ be the subgroup of $\mathrm{GL}_9(\mathbf{Z})$$\mathrm{GL}_9(\mathbf{Q})$ defined as follows: Letting $(a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8,a_9)$ be the canonical basis of $\mathbf{Z}^9$$\mathbf{Q}^9$, $G$ is generated by:

  • All permutation matrices permuting the basis elements $(a_1,\dots,a_8)$ (fixing $a_9$)
  • The additional transformation $c$ defined by $a_9\mapsto a_1+a_2$; $a_1\mapsto \frac12(a_1-a_2+a_9)$; $a_2\mapsto \frac12(a_2-a_1+a_9)$; $a_{i>2}\mapsto a_i+\frac12(a_1+a_2-a_9)$

I am interested to identify what is this group and more simply whether it is finite and if so, to determine its order.

Let $G$ be the subgroup of $\mathrm{GL}_9(\mathbf{Z})$ defined as follows: Letting $(a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8,a_9)$ be the canonical basis of $\mathbf{Z}^9$, $G$ is generated by:

  • All permutation matrices permuting the basis $(a_1,\dots,a_8)$ (fixing $a_9$)
  • The additional transformation $c$ defined by $a_9\mapsto a_1+a_2$; $a_1\mapsto \frac12(a_1-a_2+a_9)$; $a_2\mapsto \frac12(a_2-a_1+a_9)$; $a_{i>2}\mapsto a_i+\frac12(a_1+a_2-a_9)$

I am interested to identify what is this group and more simply whether it is finite and if so, to determine its order.

Let $G$ be the subgroup of $\mathrm{GL}_9(\mathbf{Q})$ defined as follows: Letting $(a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8,a_9)$ be the canonical basis of $\mathbf{Q}^9$, $G$ is generated by:

  • All permutation matrices permuting the basis elements $(a_1,\dots,a_8)$ (fixing $a_9$)
  • The additional transformation $c$ defined by $a_9\mapsto a_1+a_2$; $a_1\mapsto \frac12(a_1-a_2+a_9)$; $a_2\mapsto \frac12(a_2-a_1+a_9)$; $a_{i>2}\mapsto a_i+\frac12(a_1+a_2-a_9)$

I am interested to identify what is this group and more simply whether it is finite and if so, to determine its order.

Post Reopened by R.P., Stefan Kohl, Derek Holt, Yemon Choi, David Loeffler
Fixed grammar in the title.
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Stefan Kohl
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Identify one group of linear transformationtransformations

Clarified the question
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YCor
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What is this discrete Identify one group? of linear transformation

I have a discrete group acting on nine numbersLet $(a_9;a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8)$.$G$ be the subgroup of $\mathrm{GL}_9(\mathbf{Z})$ defined as follows: Letting $(a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8,a_9)$ be the canonical basis of $\mathbf{Z}^9$, $G$ is generated by:

  • The entries $a_1$ to $a_8$ can be permuted

    All permutation matrices permuting the basis $(a_1,\dots,a_8)$ (fixing $a_9$)
  • There is additional transformation $a_9\to a_1+a_2$; $a_1\to \frac12(a_1-a_2+a_9)$; $a_2\to \frac12(a_2-a_1+a_9)$; $a_{i>2}\to a_i+\frac12(a_1+a_2-a_9)$

    The additional transformation $c$ defined by $a_9\mapsto a_1+a_2$; $a_1\mapsto \frac12(a_1-a_2+a_9)$; $a_2\mapsto \frac12(a_2-a_1+a_9)$; $a_{i>2}\mapsto a_i+\frac12(a_1+a_2-a_9)$

I am interested to identify what is this group and more simply whatwhether it is thefinite and if so, to determine its order of this group.

What is this discrete group?

I have a discrete group acting on nine numbers $(a_9;a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8)$.

  • The entries $a_1$ to $a_8$ can be permuted

  • There is additional transformation $a_9\to a_1+a_2$; $a_1\to \frac12(a_1-a_2+a_9)$; $a_2\to \frac12(a_2-a_1+a_9)$; $a_{i>2}\to a_i+\frac12(a_1+a_2-a_9)$

I am interested to identify what is this group and more simply what is the order of this group.

Identify one group of linear transformation

Let $G$ be the subgroup of $\mathrm{GL}_9(\mathbf{Z})$ defined as follows: Letting $(a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8,a_9)$ be the canonical basis of $\mathbf{Z}^9$, $G$ is generated by:

  • All permutation matrices permuting the basis $(a_1,\dots,a_8)$ (fixing $a_9$)
  • The additional transformation $c$ defined by $a_9\mapsto a_1+a_2$; $a_1\mapsto \frac12(a_1-a_2+a_9)$; $a_2\mapsto \frac12(a_2-a_1+a_9)$; $a_{i>2}\mapsto a_i+\frac12(a_1+a_2-a_9)$

I am interested to identify what is this group and more simply whether it is finite and if so, to determine its order.

Post Closed as "Needs details or clarity" by Derek Holt, user6976, Neil Strickland, Sebastian Goette, Alain Valette
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