Dimension subgroups, nilpotency indices, and the number of generators of ideals in p-group algebras.

*(English)*Zbl 0695.16007Let K be a field of non-zero characteristic p. Let G be a finite p-group of order \(p^ n\). Then the augmentation ideal \(\Delta\) of the group algebra KG is nilpotent, and its index of nilpotence is denoted by t(G). It is well known that \(t(G)\leq p^ n\), with equality if and only if G is cyclic. S. Koshitani [Tsukuba J. Math. 1, 137-148 (1977; Zbl 0391.16007)] has proved that if G is not cyclic then \(t(G)\leq p^{n-1}- p+1\), with equality if and only if G contains a cyclic subgroup of index p. Now suppose that \(p\neq 2\). It is proved that for any \(i>0\) and sufficiently large n, \(t(G)\geq p^{n-i}\) if and only if G contains a cyclic subgroup of index \(p^ i\). Moreover, if \(\exp (G)=p^ e\), then \(t(G)\leq p^ e+p^{n-e}-1\), with equality if and only if G is metacyclic.

The Loewy series \(\{c_ i\}\) is defined by \(c_ i=\dim (\Delta^ i/\Delta^{i+1})\). It is known that \(\{c_ i\}\) is not unimodal in general. It is proved that the Loewy series is unimodal if \(n\leq 2e\), and this generalizes a result of O. Manz and R. Staszewski for metacyclic groups [see their papers in Proc. Am. Math. Soc. 98, 189-195 (1986; Zbl 0611.20003) and J. Reine Angew. Math. 368, 108-118 (1986; Zbl 0586.20006)].

Let v(KG) denote the least upper bound of the minimal numbers of generators of left ideals in KG. It is not difficult to establish that max \(c_ i\leq v(KG)\leq p^{n-e}\). It is proved that \(v(KG)=p^{n-e}\) provided 2e\(\geq n\). Moreover, if G is sufficiently large, then \(v(KG)\leq p^ i\) if and only if G contains a cyclic subgroup of index \(p^ i.\)

Infinite groups are discussed briefly. Let G be a residually-p group. Then v(KG) is finite if and only if G has a cyclic subgroup of finite index, and in this case \(v(KG)=\min \{[G:C]:\) C is a cyclic subgroup of \(G\}\) and v(KG) is a power of p. All of these results are consequences of a detailed study of the dimension subgroups of KG.

The Loewy series \(\{c_ i\}\) is defined by \(c_ i=\dim (\Delta^ i/\Delta^{i+1})\). It is known that \(\{c_ i\}\) is not unimodal in general. It is proved that the Loewy series is unimodal if \(n\leq 2e\), and this generalizes a result of O. Manz and R. Staszewski for metacyclic groups [see their papers in Proc. Am. Math. Soc. 98, 189-195 (1986; Zbl 0611.20003) and J. Reine Angew. Math. 368, 108-118 (1986; Zbl 0586.20006)].

Let v(KG) denote the least upper bound of the minimal numbers of generators of left ideals in KG. It is not difficult to establish that max \(c_ i\leq v(KG)\leq p^{n-e}\). It is proved that \(v(KG)=p^{n-e}\) provided 2e\(\geq n\). Moreover, if G is sufficiently large, then \(v(KG)\leq p^ i\) if and only if G contains a cyclic subgroup of index \(p^ i.\)

Infinite groups are discussed briefly. Let G be a residually-p group. Then v(KG) is finite if and only if G has a cyclic subgroup of finite index, and in this case \(v(KG)=\min \{[G:C]:\) C is a cyclic subgroup of \(G\}\) and v(KG) is a power of p. All of these results are consequences of a detailed study of the dimension subgroups of KG.

Reviewer: P.F.Smith

##### MSC:

16S34 | Group rings |

20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |

20D15 | Finite nilpotent groups, \(p\)-groups |

16N40 | Nil and nilpotent radicals, sets, ideals, associative rings |

16Dxx | Modules, bimodules and ideals in associative algebras |

##### Keywords:

finite p-group; augmentation ideal; group algebra; index of nilpotence; cyclic subgroup; Loewy series; metacyclic groups; minimal numbers of generators; residually-p group; cyclic subgroup of finite index; dimension subgroups
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