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)].

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)].

Reviewer: Klaus Roggenkamp (Stuttgart)