Centrale-Supélec 2026 — Mathématiques 1 (MP, MPI)

Independent solutions, written by Pulsars. Neither official nor affiliated with Concours Centrale-Supélec.

1. Exercise 1 — Partie A · I — Généralités (Q1 à Q3)

   G denotes a subgroup of GL_n(R). A group is said to have finite exponent when a single integer m satisfies x^m = e for every element, and we denote by tr G the set of traces of its elements.

   • Q1 — a finite subgroup has finite exponent and its trace set is finite.
   • Q2 — a counterexample: an infinite subgroup with a finite trace set.
   • Q3 — an infinite group of finite exponent, of matrices or not.
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2. Exercise 2 — Partie A · II — Les sous-groupes finis de O₂(R) (Q4 à Q8)

   We denote by R_θ the rotation of angle θ and by S_θ the associated reflection, with the relations recalled in the paper.

   • Q4 — every matrix of SO₂(R) can be written R_θ for a unique θ in [0, 2π[.
   • Q5 — study of the group generated by a single rotation: form of the elements, cardinality for θ = 2π/m, finiteness equivalent to θ/π rational, and the two converses (finite exponent, finite trace set).
   • Q6 — the group generated by two rotations, then the case ⟨R_{π/p}, R_{π/q}⟩.
   • Q7 — every finite subgroup of SO₂(R) is cyclic.
   • Q8 — the dihedral group D_m: its cardinality, and the fact that every finite subgroup of O₂(R) not contained in SO₂(R) is isomorphic to it.
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3. Exercise 3 — Partie A · III — Caractérisation des sous-groupes finis de Oₙ(R) (Q9 à Q14)

   G is here a subgroup of O_n(R), and we want to establish the equivalences

   G fini ⟺ G d'exposant fini ⟺ tr G fini.

   • Q9 — recall the reduction theorem for orthogonal matrices.
   • Q10 — finite exponent implies finite trace set.
   • Q11 to Q14 — the converse, by considering the subspace F spanned by G in M_n(R), a basis extracted from G, the associated Gram matrix and the coordinates of an element of G in this basis.
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4. Exercise 4 — Partie B · I — Quelques propriétés utiles (Q15 à Q21)

   A and B denote symmetric matrices, A positive definite. We set ⟨X, Y⟩_A = (AX|Y) and write B ≼ A when (BX|X) ⩽ (AX|X) for every X.

   • Q15 — ⟨.,.⟩_A is an inner product; is a Euclidean ball a ball for such a norm?
   • Q16 — X ↦ A^{-1}BX is self-adjoint for this inner product.
   • Q17 to Q21 — characterisations of ≼: by the positivity of A − B, by the spectrum of A^{-1}B, by determinants, by inclusion of balls, and finally the link between a bounded subset of S_n^+(R) and being bounded above by αI_n.
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5. Exercise 5 — Partie B · II — Stricte log-concavité du déterminant (Q22, Q23)

   • Q22 — S_n^{++}(R) is a convex subset of M_n(R).
   • Q23 — A ↦ ln(det A) is strictly concave on it, in the sense defined by the paper (equality only if A = B, t = 0 or t = 1).
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6. Exercise 6 — Partie B · III — Groupe orthogonal d'une matrice définie positive (Q24, Q25)

   To A ∈ S_n^{++}(R) we associate O(A) = {M ∈ M_n(R) ; M^T A M = A}.

   • Q24 — O(A) is a subgroup of GL_n(R), isomorphic to O_n(R).
   • Q25 — O(A) is a compact subset of M_n(R).
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7. Exercise 7 — Partie B · IV — Les sous-groupes compacts de GLₙ(R) (Q26 à Q29)

   G denotes a compact subgroup of GL_n(R). The aim is to show that it is isomorphic to a subgroup of O_n(R).

   • Q26 — every element of G has a determinant of absolute value 1.
   • Q27 — the set C = {MX ; M ∈ G, X ∈ B} is compact and 0 is an interior point of it.
   • Q28 — there exists a unique positive definite matrix A of maximal determinant such that C ⊂ B_A.
   • Q29 — G is then a subgroup of O(A).
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8. Exercise 8 — Partie C · I — Croissance d'un groupe (Q30 à Q32)

   G is a group generated by a finite subset S, and ℓ_S(g) denotes the minimal length of an expression of g in terms of the generators. V_S(p) is the ball of radius p for this length.

   • Q30 — ℓ_S(g^{-1}) = ℓ_S(g) and ℓ_S(gh) ⩽ ℓ_S(g) + ℓ_S(h).
   • Q31 — two finite generating sets give comparable lengths: the degree of polynomial growth does not depend on the chosen system.
   • Q32 — counting triples of natural numbers with bounded sum, and degree of growth of (Z³, +).
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9. Exercise 9 — Partie C · II — Le groupe de Heisenberg discret (Q33 à Q41)

   n = 3. S, T, U are the three unipotent matrices of the statement, A the set they form, and H the subgroup of GL₃(R) they generate.

   • Q33 to Q37 — general form of S^i T^j U^k, composition law, centrality of U, non-commutativity of H, and the bijection Z³ → H.
   • Q38 to Q40 — upper and lower bounds for the cardinality of V_A(p), with the associated count.
   • Q41 — conclusion: H has polynomial growth of degree 4.
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