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

1 →H2(G/N,A) → H2(G, A) →H2(N,A)
when Hom(N, A) = 1 .

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) →H2 (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→H2 (T , N* ) → M˜(G) → M(N)T→H2 (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.