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Factorization numbers of a class of finite \(p\)-groups. (English) Zbl 1438.20027

Summary: Let \(p\) be a prime number and \({f_2} (G)\) be the number of factorizations \(G = AB\) of the group \(G\), where \(A\), \(B\) are subgroups of \(G\). Let \(G\) be a class of finite \(p\)-groups as follows, \(G = \langle a, b|a^{p^n} = b^{p^m} = 1, a^b = a^{p^{n-1}+1}\rangle\), where \(n > m \ge 1\). In this article, the factorization number \({f_2} (G)\) of \(G\) is computed, which improves the results in literature.

MSC:

20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D40 Products of subgroups of abstract finite groups
20D15 Finite nilpotent groups, \(p\)-groups
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