Classification of finite p-groups of characteristic rank one

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 $G$ 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 $2^{r-2}+1$.

Related facts

• Classification of finite p-groups of normal rank one
• Classification of finite p-groups of rank one

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 198-199, Theorem 4.9, Section 5.4 ($p$-groups of small depth)