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Finite group, N_{ABC} = N_{CBA}

Source: IMC 2007, Day 1, Problem 4

August 6, 2007

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Problem Statement

Let

G

be a finite group. For arbitrary sets

U, V, W \subset G

, denote by

N_{UVW}

the number of triples

(x, y, z) \in U \times V \times W

for which

xyz

is the unity . Suppose that

G

is partitioned into three sets

N_{ABC}= N_{CBA}.

and

C

(i.e. sets

A, B, C

are pairwise disjoint and

G = A \cup B \cup C

). Prove that

N_{ABC}= N_{CBA}.

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