On Sylow \(p\)-subgroups of general linear groups over commutative local rings of characteristic \(p^s\) (Q2759420)

The main result of the article is the following. Let \(K\) be a commutative ring with unit of characteristic \(p^s\) (\(s>0\), \(p\) a prime), let \(n\) be a positive integer, and let \(\phi\colon\text{GL}(n,K)\to\text{GL}(n,K/\text{rad }K)\) be a homomorphism of the group \(\text{GL}(n,K)\) onto the group \(GL(n,K/\text{rad }K)\). Under this mapping the inverse image of any Sylow \(p\)-subgroup of the group \(\text{GL}(n,K/\text{rad }K)\) is a Sylow \(p\)-subgroup of the group \(\text{GL}(n,K)\) and conversely, any Sylow \(p\)-subgroup of the group \(\text{GL}(n,K)\) is the inverse image of some Sylow \(p\)-subgroup of the group \(\text{GL}(n,K/\text{rad K})\). Two Sylow \(p\)-subgroups of the group \(\text{GL}(n,K)\) are conjugate if and only if their images under the homomorphism \(\phi\) are conjugate Sylow \(p\)-subgroups of the group \(GL(n,K/\text{rad }K)\). A description of a Sylow \(p\)-subgroup of a general linear group over a commutative ring of characteristic \(p^s\) is also given.