Classification of finite p-groups of characteristic rank one

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This article gives a classification statement for certain kinds of groups of prime power order, subject to additional constraints.
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Suppose G is a group of prime power order that is also of characteristic rank one. Then, G is expressible asa central product ER, such that:

  • E is either trivial or extraspecial
  • R is a cyclic group if p is an odd prime. For p=2, R is either cyclic, or has a cyclic maximal subgroup with the quotient acting on it by multiplication by either -1 or 2^{r-2} - 1, where |R| = 2^r. In particular, it has a cyclic maximal subgroup but is not of the form where the quotient acts via multiplication by 2^{r-2} + 1.

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