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On products of cyclic and abelian finite \(p\)-groups (\( p\) odd). (English) Zbl 1515.20114

Summary: For an odd prime \(p\), it is shown that if \(G = AB\) is a finite \(p\)-group, for subgroups \(A\) and \(B\) such that \(A\) is cyclic and \(B\) is abelian of exponent at most \(p^{k}\), then \(\Omega_{k}(A)B \trianglelefteq G\), where \(\Omega_{k}(A) = \langle g \in A \mid g^{ p^{k}} = 1 \rangle\).

MSC:

20D40 Products of subgroups of abstract finite groups
20D15 Finite nilpotent groups, \(p\)-groups
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References:

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