Classification of finite p-groups of normal 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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Statement

Suppose $P$ is a finite $p$-group whose normal rank is one: every Abelian normal subgroup of $P$ is cyclic. Then:

• If $p$ is odd, $P$ is itself cyclic.
• If $p=2$, $P$ is either cyclic, or it has a cyclic maximal subgroup, with the quotient acting by multiplication by either $-1$ or $2^{r-2}-1$, where $\left|P\right|=2^r$.

Related facts

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

Facts used

1. Classification of finite p-groups of characteristic rank one
2. Extraspecial and rank one implies quaternion group

References

Textbook references

• Finite Groups by Daniel Gorenstein, ISBN 0821843427, ^{More info}, Page 199, Theorem 4.10(i), Section 5.4 ($p$-groups of small depth)