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4

Part of 1993 Hungary-Israel Binational

Problems(2)

rooks on a 3n x 3n chessboard

Source: 4-th Hungary-Israel Binational Mathematical Competition 1993

5/26/2007
Find the largest possible number of rooks that can be placed on a 3n×3n3n \times 3n chessboard so that each rook is attacked by at most one rook.
combinatorics unsolvedcombinatorics
|aH\cap Hb| and |H|, here H\leq G

Source: 4-th Hungary-Israel Binational Mathematical Competition 1993

5/27/2007
In the questions below: GG is a finite group; H≤GH \leq G a subgroup of G;∣G:H∣G; |G : H | the index of HH in G;∣X∣G; |X | the number of elements of X⊆G;Z(G)X \subseteq G; Z (G) the center of G;G′G; G' the commutator subgroup of G;NG(H)G; N_{G}(H ) the normalizer of HH in G;CG(H)G; C_{G}(H ) the centralizer of HH in GG; and SnS_{n} the nn-th symmetric group. Let H≤GH \leq G and a,b∈G.a, b \in G. Prove that ∣aH∩Hb∣|aH \cap Hb| is either zero or a divisor of ∣H∣.|H |.
group theoryabstract algebrasuperior algebrasuperior algebra unsolved