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Publication# Elementary abelian subgroups in p-groups of class 2

Abstract

All the results in this work concern (finite) p-groups. Chapter 1 is concerned with classifications of some classes of p-groups of class 2 and there are no particularly new results in this chapter, which serves more as an introductory chapter. The "geometric" method we use for these classifications differs however from the standard approach, especially for p-groups of class 2 with cyclic center, and can be of some interest in this situation. This "geometry" will for instance, prove to be particularly useful for the description of the automorphism groups performed in Chapter 3. Our main results can be found in chapters 2 and Chapter 3. The results of Chapter 2 have a geometric flavour and concern the study of upper intervals in the poset Ap(P) for p-groups P. We already know from work of Bouc and Thévenaz [8], that Ap(P)≥2 is always homotopy equivalent to a wedge of spheres. The first main result in Section 2.4, is a sharp upper bound, depending only on the order of the group, to the dimension of the spheres occurring in Ap(P)≥2. More precisely, we show that if P has order pn, then H~k(Ap(P)≥2) = 0 if k ≥ ⎣n-1/2⎦. The second main result in this section is a characterization of the p-groups for which this bound is reached. The main results in Section 2.3 are numerical values for the number of the spheres occurring in Ap(P)≥2 and their dimension, when P is a p-group with a cyclic derived subgroup. Using these calculations, we determine precisely in Section 2.5, for which p-groups with a cyclic center, the poset Ap(P) is homotopically Cohen-Macaulay. Section 2.7 is an attempt to generalize the work of Bouc and Thévenaz [8]. The main result of this section is a spectral sequence E1rs converging to H~r+s(Ap(P)>Z), for any Z ∈ Ap(P). We show also that this spectral sequence can be used to recover Bouc and Thévenaz's results [8]. In Section 2.8, we give counterexamples to results of Fumagalli [12]. As an important consequence, Fumagalli's claim that Ap(G) is homotopy equivalent to a wedge of spheres, for solvable groups G, seems to remain an open question. The results of Chapter 3 are more algebraic and concern automorphism groups of p-groups. The main result is a description of Aut(P), when P is any group in one of the following two classes: p-groups with a cyclic Frattini subgroup. odd order p-groups of class 2 such that the quotient by the center is homocyclic.

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Related concepts (43)

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Ontological neighbourhood

P-group

In mathematics, specifically group theory, given a prime number p, a p-group is a group in which the order of every element is a power of p. That is, for each element g of a p-group G, there exists a nonnegative integer n such that the product of pn copies of g, and not fewer, is equal to the identity element. The orders of different elements may be different powers of p. Abelian p-groups are also called p-primary or simply primary. A finite group is a p-group if and only if its order (the number of its elements) is a power of p.

Solvable group

In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a solvable group is a group whose derived series terminates in the trivial subgroup. Historically, the word "solvable" arose from Galois theory and the proof of the general unsolvability of quintic equation. Specifically, a polynomial equation is solvable in radicals if and only if the corresponding Galois group is solvable (note this theorem holds only in characteristic 0).

Prüfer group

In mathematics, specifically in group theory, the Prüfer p-group or the p-quasicyclic group or p∞-group, Z(p∞), for a prime number p is the unique p-group in which every element has p different p-th roots. The Prüfer p-groups are countable abelian groups that are important in the classification of infinite abelian groups: they (along with the group of rational numbers) form the smallest building blocks of all divisible groups. The groups are named after Heinz Prüfer, a German mathematician of the early 20th century.

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