Order statistics-equivalence is finite-direct product-closed

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Statement

Statement for direct product of two groups

Suppose G_1, G_2, H_1, H_2 are finite groups such that G_1 is order statistics-equivalent to H_1 and G_2 is order statistics-equivalent to H_2. Then, the External direct product (?) G_1 \times G_2 is order statistics-equivalent to the external direct product H_1 \times H_2.

Here, two finite groups are termed order statistics-equivalent finite groups if they have the same order statistics, i.e., the same number of elements of each order.

Statement for direct product of finitely many groups

Suppose G_1, G_2, \dots, G_n and H_1, H_2, \dots, H_n are groups such that, for each i satisfying 1 \le i \le n, G_i is order statistics-equivalent to H_i. Then, the External direct product (?) G_1 \times G_2 \times \dots \times G_n is order statistics-equivalent to H_1 \times H_2 \times \dots \times H_n.

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