Prime-power lie algebras and finite p-groups

In this thesis we use the Lie ring functors of Magnus and Lazard to investigate finite p-groups which possess either a cyclic subgroup of small index, or whose derived subgroup has exponent dividing p. The class of groups we consider is sufficiently large to include completely 11 of the 15 families of groups of order p7 (p > 7), partitioned by Hall’s type invariants. Some information on the other 4 families is also derived. Motivated by the work of Burnside [3] and Miller [21] concerning the stable behaviour of groups of order pn and exponent pn-2 for p and n sufficiently large, we investigate the existence of stability for the number of isomorphism classes of groups of order pn and exponent pn-f as p and n vary, with f an arbitrary fixed integer greater than 2. To facilitate this, we allow finitely many specified primes to be excluded for each / at the outset. The approach is to then consider the corresponding problem for pn—element nilpotent Lie rings and use the Lie ring functors to recover a solution for the groups. A non-trivial step immediately arises in showing the existence of an initial set of excluded primes which are sufficient to enable the Lie ring functors to be invoked since they only apply when the prime is greater than the nilpotency class (of both the groups and Lie rings). This step is dealt with and explored in chapters 2 and 3. In the main theorem of chapter 4 we show that for f > 3 the number of isomorphism classes of groups of order pn and exponent pn-f is independent of n for n sufficiently large and p not one of the excluded primes (which depend only on f). The excluded primes ensure regularity holds for such groups and the proof of this theorem yields precise stability results in terms of the corresponding type invariants. The method of proof shows explicitly how to construct the corresponding Lie rings, and in chapter 5 we utilise this procedure to produce a formula for the number of groups of order pn and exponent p"—3 where p > 5 and n > 7. The precise stability results of chapter 4 enable us to reduce some of the calculations to the known classification of groups of order p5 (p > 5). On the other hand, in chapter 6 we use the Lie ring functors to solve a restricted form of a conjecture of J. Moody [22] by exhibiting, for a prime p greater than or equal to the positive integer n, a natural, but not functorial, one-to-one correspondence between isomorphism classes of finite groups of order pn whose derived subgroup has exponent dividing p, and isomorphism classes of nilpotent Fp[T]/(T")—Lie algebras L of Fp—dimension n in which T[L,L] = 0. By viewing such an algebra as a nilpotent Fp —Lie algebra equipped with a nilpotent element of its centroid one obtains a “formula” for the number of such groups. This applies, in particular, to the groups of order p7 since the 7-dimensional nilpotent Fp —Lie algebras are known from [30] (and the Magnus Lazard functors) for p > 7. In view of current interest in these groups, we conclude with a summary of the information contained in this thesis on groups of order p7

Derived induction and restriction theory

Let $G$ be a finite group. To any family $\mathscr{F}$ of subgroups of $G$, we associate a thick $\otimes$-ideal $\mathscr{F}^{\mathrm{Nil}}$ of the category of $G$-spectra with the property that every $G$-spectrum in $\mathscr{F}^{\mathrm{Nil}}$ (which we call $\mathscr{F}$-nilpotent) can be reconstructed from its underlying $H$-spectra as $H$ varies over $\mathscr{F}$. A similar result holds for calculating $G$-equivariant homotopy classes of maps into such spectra via an appropriate homotopy limit spectral sequence. In general, the condition $E\in \mathscr{F}^{\mathrm{Nil}}$ implies strong collapse results for this spectral sequence as well as its dual homotopy colimit spectral sequence. As applications, we obtain Artin and Brauer type induction theorems for $G$-equivariant $E$-homology and cohomology, and generalizations of Quillen's $\mathcal{F}_p$-isomorphism theorem when $E$ is a homotopy commutative $G$-ring spectrum. We show that the subcategory $\mathscr{F}^{\mathrm{Nil}}$ contains many $G$-spectra of interest for relatively small families $\mathscr{F}$. These include $G$-equivariant real and complex $K$-theory as well as the Borel-equivariant cohomology theories associated to complex oriented ring spectra, any $L_n$-local spectrum, the classical bordism theories, connective real $K$-theory, and any of the standard variants of topological modular forms. In each of these cases we identify the minimal family such that these results hold.Comment: 63 pages. Many edits and some simplifications. Final version, to appear in Geometry and Topolog