Syllabus:
1) Basic set-theoretic terminology, equivalence relations, quotients.
2) Groups, subgroups, quotient of a group by a subgroup, normal subgroups. (...)
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1) Basic set-theoretic terminology, equivalence relations, quotients.
2) Groups, subgroups, quotient of a group by a subgroup, normal subgroups.
Basic examples: permuation groups, linear groups.
Rings, fields (we shall only give the definitions and some examples, without systematic investigation).
3) Properties of Z, elementary arithmetic, finitely generated abelian groups.
4) Group action on a set, symmetric group. Application: Sylow's theorems.
5) Free groups, groups presented with generators and relations.
6) Exact sequences, semi-direct product, dévissages of groups (Jordan-Hölder, solvable groups, nilpotent groups...).
7) Reminders of linear algebra. Basic notions in representation theory; sub-representations, quotient representations, induced representations, linearization of a set-theoretic action...
Irreducible and indecomposable representation, Schur's Lemma, existence of uniqueness of a decomposition as a direct sum of irreducible representations (under suitable assumptions).
Character of a representation.
8) Tensor product of vector spaces, and of representations.
9) Complex representations, orthogonality of characters, construction of character tables.
10) Complex representations of the symmetric group.
11) Bilinear form; the symmetric case, the alternate case. Quadratic forms, Witt's theorem. Sesquilinear forms over C.
12) Study of the subgroups of GLn associated tio bilinear forms: orthogonal group, symplectic group, unitary group...