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A well-known result for Vilenkin systems is the fact that for all 1 < p < ∞ the n-th partial sums of Fourier series of all functions in the space Lp converge to the function in Lp-norm. This statement can not be generalized to any representative product system on the complete product of finite non-abelian groups, but even then it is true for the complete product of quaternion groups with bounded orders and monomial representative product system ordered in a specific way.