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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Definition
Suppose is a group of prime power order that is also of characteristic rank one. Then, is expressible asa central product , such that:
- is either trivial or extraspecial
- is a cyclic group if is an odd prime. For , is either cyclic, or has a cyclic maximal subgroup with the quotient acting on it by multiplication by either or , where . In particular, it has a cyclic maximal subgroup but is not of the form where the quotient acts via multiplication by .
Related facts
References
Textbook references
- Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 198-199, Theorem 4.9, Section 5.4 (-groups of small depth)