# Syllabus

**Schur Multiplier (Cohomological Approach):**Cohomolgy Groups, second cohomology group and central extensions, the homomorphisms restriction, inflation and transgression, Hochschild–Serre exact sequence, some extensions of the 5–term sequence, second cohomology group for a cyclic group of finite order, the homomorphisms conjugation, corestriction and connection between restriction and corestriction maps, the exact sequence

^{2}(G/N,A) → H

^{2}(G, A) →H

^{2}(N,A)

Elementary results of Schur (1904) on the multiplicator M(G), result of Alperin and Tze–Nan Kuo connecting the exponents of M(G), G and the order of G. The exact sequence

1→ Ext(G/G′,A) →H

^{2}(G,A) →Hom(A*,M(G))→1, where A*= Hom(A, C*).

Existence of a covering group G* of G ≅ H ⁄A when A is a subgroup of H’ ∩ Z (H ). The exact sequence 1→H

^{2}(T , N* ) → M

^{˜}(G) → M(N)

^{T}→H

^{2}(T, N*) when G is the semidirect product of a normal subgroup N and a subgroup T, where M

^{˜}(G) is the kernel of the restriction map res: M(G)→M(T).

**Schur Multiplier (Combinatorial Approach ) :**Hopf’s formula, covering groups and non-abelian exterior squares of groups, a survey of bounds on the order, rank, and exponent of the Schur multiplier, algorithmic methods for computing Schur multipliers, some obstructions for universal commutator relations of groups.

**Tensor Product of Groups:**Compatibility conditions, non-abelian tensor square and non-abelian tensor product, some examples of actions and basic properties, proof of finiteness of non-abelian tensor product if both factors are finite, some other types of tensor product including Box tensor product.