Classification of finite p-groups of 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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Statement

Let p be a prime, and let P be a finite p-group (a group whose order is a power of p), such that P has rank at most one: in other words, every Abelian subgroup of P is cyclic. Then:

  1. If p is odd, then P must be cyclic
  2. If p = 2, then P is either cyclic or is a generalized quaternion group

Facts used

  1. Classification of finite p-groups of normal rank one: For an odd prime p, any finite p-group of normal rank one is cyclic. For p = 2, it is either cyclic or dihedral or generalized quaternion, or a semidirect product of a cyclic maximal subgroup with a two-element subgroup acting via multiplication by 2^{r-2} - 1 where the order of the group is 2^r.

Proof

If P is a finite p-group of rank one, then it in particular has normal rank one. Thus, fact (1) applies, and we get:

  • For odd primes p, we have the result.
  • For p = 2, we have the cyclic and generalized quaternion groups, as well as two other kinds of groups. Both these other kinds of groups, however, possess elements of order two outside the center, and the subgroups generated by these elements and the center are Abelian and not cyclic. Thus, the cyclic and generalized quaternion groups are the only groups left.

References

Textbook references

  • Finite Groups by Daniel Gorenstein, ISBN 0821843427, More info, Page 199, Theorem 4.10(ii), Section 5.4 (p'-automorphisms of p-groups)