grandes-ecoles 2014 QIII.A.1

grandes-ecoles · France · centrale-maths2__mp Groups Group Homomorphisms and Isomorphisms

We equip $\mathbb{C}[X]$ with the internal composition law given by composition, denoted $\circ$. We seek families $(F_n)_{n \in \mathbb{N}}$ of complex polynomials satisfying $$\forall n \in \mathbb{N}, \quad \deg F_n = n \quad \text{and} \quad \forall (m,n) \in \mathbb{N}^2, \quad F_n \circ F_m = F_m \circ F_n \tag{III.1}$$
Show that the family $(T_n)_{n \in \mathbb{N}}$ satisfies property (III.1). One may compare $T_n \circ T_m$ and $T_{mn}$.

We equip $\mathbb{C}[X]$ with the internal composition law given by composition, denoted $\circ$. We seek families $(F_n)_{n \in \mathbb{N}}$ of complex polynomials satisfying
$$\forall n \in \mathbb{N}, \quad \deg F_n = n \quad \text{and} \quad \forall (m,n) \in \mathbb{N}^2, \quad F_n \circ F_m = F_m \circ F_n \tag{III.1}$$

Show that the family $(T_n)_{n \in \mathbb{N}}$ satisfies property (III.1). One may compare $T_n \circ T_m$ and $T_{mn}$.

Paper Questions

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