On linear just infinite pro-\(p\) groups. (English) Zbl 1018.20022
The author studies linearity of pro-\(p\) groups over commutative rings and over commutative profinite rings, in particular linearity of just infinite pro-\(p\) groups, i.e. groups whose proper quotients are finite. A group \(G\) is called hereditarily just infinite if every proper quotient of any subgroup of finite index of \(G\) is finite. Among others the following results are proved.
Proposition 1.1. Let \(G\) be a just infinite pro-\(p\) group. Then \(G\) is insoluble if and only if every non-trivial normal (not necessarely closed) subgroup of \(G\) is open.
Corollary 1.2. Let \(G\) be an insoluble hereditarily just infinite pro-\(p\) group. Then \(G\) is hereditarily just infinite as an abstract group.
Corollary 1.5. Let \(G\) be a linear just infinite pro-\(p\) group over some commutative profinite ring. Then \(G\) is linear either over the \(p\)-adic integers \(\mathbb{Z}_p\) or over the power series ring \(\mathbb{F}_p[[t]]\), where \(\mathbb{F}_p\) is the field of \(p\) elements.
Theorem 1.8. Let \(G\) be a \(p\)-adic analytic pro-\(p\) group. Then \(G\) is linear over a field of positive characteristic if and only if \(G\) is virtually Abelian.
Proposition 1.1. Let \(G\) be a just infinite pro-\(p\) group. Then \(G\) is insoluble if and only if every non-trivial normal (not necessarely closed) subgroup of \(G\) is open.
Corollary 1.2. Let \(G\) be an insoluble hereditarily just infinite pro-\(p\) group. Then \(G\) is hereditarily just infinite as an abstract group.
Corollary 1.5. Let \(G\) be a linear just infinite pro-\(p\) group over some commutative profinite ring. Then \(G\) is linear either over the \(p\)-adic integers \(\mathbb{Z}_p\) or over the power series ring \(\mathbb{F}_p[[t]]\), where \(\mathbb{F}_p\) is the field of \(p\) elements.
Theorem 1.8. Let \(G\) be a \(p\)-adic analytic pro-\(p\) group. Then \(G\) is linear over a field of positive characteristic if and only if \(G\) is virtually Abelian.
Reviewer: Pavel Zalesskij (Brasilia)
MSC:
| 20E18 | Limits, profinite groups |
| 20G25 | Linear algebraic groups over local fields and their integers |
Keywords:
just infinite pro-\(p\) groups; linear pro-\(p\) groups; subgroups of finite index; hereditarily just infinite groupsReferences:
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