grandes-ecoles 2015 QV.A.1

grandes-ecoles · France · centrale-maths1__mp Groups Group Actions and Surjectivity/Injectivity of Maps

If $f$ is a function defined on $\mathbb{R}^2$, we denote by $f^*$ the function $f \circ \Phi$, defined on $G$ by $f^*(g) = f(\Phi(g))$ where $\Phi : G \rightarrow \mathbb{R}^2$ is the function introduced in question I.A.5.
Prove that for all $g$ in $G$ and $r$ such that $\Phi(r) = \overrightarrow{0}$ we have $f^*(gr) = f^*(g)$.

If $f$ is a function defined on $\mathbb{R}^2$, we denote by $f^*$ the function $f \circ \Phi$, defined on $G$ by $f^*(g) = f(\Phi(g))$ where $\Phi : G \rightarrow \mathbb{R}^2$ is the function introduced in question I.A.5.

Prove that for all $g$ in $G$ and $r$ such that $\Phi(r) = \overrightarrow{0}$ we have $f^*(gr) = f^*(g)$.

Paper Questions

QI.A.1 QI.A.2 QI.A.3 QI.A.4 QI.A.5 QI.B.1 QI.B.2 QI.B.3 QI.B.4 QI.C.1 QI.C.2 QI.C.3 QI.C.4 QII.A.1 QII.A.2 QII.A.3 QII.A.4 QII.B.1 QII.B.2 QII.B.3 QII.B.4 QIII.A QIII.B QIII.C.1 QIII.C.2 QIII.C.3 QIV.A.1 QIV.A.2 QIV.B.1 QIV.B.2 QIV.B.3 QIV.C.1 QIV.C.2 QIV.C.3 QIV.D.1 QIV.D.2 QIV.D.3 QV.A.1 QV.A.2 QV.A.3 QV.B.1 QV.B.2