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Groups of non-zero presentation rank. (English) Zbl 0358.20045
Symp. math. 17, Anelli Funz. contin., Gruppi infin., Convegni 1973, 215-224 (1976).
[This article was published in the book announced in this Zbl 0317.00007.]
Let $$G$$ be a finite group with minimal number of free generators $$d = d(G)$$, and let $$F$$ be free on $$d$$ elements mapping onto $$G$$, with kernel $$R$$, the quotient $$\bar{R}=R/[R,R]$$ is called a relation module of $$G$$, with $$G$$ acting via conjugation. Decompose $$\bar{R}$$ as $$\bar{R}=A \oplus P$$, where $$P$$ is projective and $$A$$ has no projective direct summand. Then $$\mathbb{Q} P \cong \mathbb{Q} G^{(r)}$$, and $$r$$ is an invariant of $$G$$ called the presentation rank of $$G$$, $$\mathrm{pr}(G)$$. This invariant strongly interrelates group theory and integral representation theory; for on the one hand we have $$\mathrm{pr}(G)>0$$ if $$d(C_{\infty}\wr G)=d(G)$$, where $$C_{\infty}$$ is the infinite cyclic group; on the other hand $$\mathrm{pr}(G)=d(G)-d_{\mathbb{Z} G}(\mathfrak{g})$$, where $$\mathfrak{g}$$ is the integral augmentation ideal and $$d_{\mathbb{Z} G}(\mathfrak{g})$$ denotes the minimal number of generators of $$\mathfrak{g}$$ as $$\mathbb{Z} G$$-module. The main tool to calculate $$\mathrm{pr}(G)$$ is the following: $$\mathrm{pr}(G)>0$$ if for all $$p\mid|G|$$ and all irreducible $$\mathbb{F}_{p}$$-modules $$M$$, $$|H^{1}(G,M)\leq |M|^{d(G)-1-\zeta_{M}}$$ where $$\zeta_{M}$$ is 1 or 0 according as $$M$$ is not or is $$\mathbb{F}_{p}$$. This result is used to prove among others the following interesting results: If $$G$$ is soluble or $$d(G)\leq2$$, then $$\mathrm{pr}(G)=0$$. On the other hand, let $$N\vartriangleleft G$$, $$d(N)\leq d(G)$$ and $$N/[N,N] \subseteq \mathrm{Frat}(G/[N,N])$$. Then $$\mathrm{pr}(G) >0$$ if (i) $$d(N) <d(G)$$ or $$\mathrm{pr}(N) >0$$ and (ii) $$d(G/N)<d(G)$$ or $$\mathrm{pr}(G/N)>0$$. This can be used to show that $$\mathrm{pr}(k \times G)>0$$ for all sufficiently large $$k$$ if and only if $$G=[G,G]$$. The smallest example is $$k \times A_{5}$$ - a rather large group. For more results in this connection we refer to K. W. Gruenberg [“Relationmodules of finite groups”, AMS Regional Conference Ser. Math. 25 (1974); see also Math. Z. 118, 30–33 (1970; Zbl 0213.30303)].

MSC:
 20F05 Generators, relations, and presentations of groups 20F40 Associated Lie structures for groups