Let ℙ be the set of all prime numbers. A subgroup H of a finite group G is called ℙ-subnormal if either H = G or there exists a chain of subgroups H = H₀ ≤ H₁ ≤ … ≤ H_n = G such that |H_i : H_i − 1| ∈ ℙ, 1 ≤ i ≤ n. We prove that any finite group with ℙ-subnormal Sylow p-subgroup of odd order is p-solvable and any group with ℙ-subnormal generalized Schmidt subgroups is metanilpotent.

中文翻译：

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ℙ次正规Sylow子群的有限群

令ℙ为所有质数的集合。一个分组ħ有限群G ^称为ℙ-低于正常如果任ħ = ģ或存在子群的链ħ  =  ħ ₀  ≤  ħ ₁  ≤...≤  ħ _Ñ  =  g ^使得| H _i  ：H _{i  − 1} | ∈ℙ，1≤ 我 ≤  Ñ。我们证明任何具有ℙ-次正规Sylow p-奇数次子组的有限群是p-可解决的，且任何具有ℙ-次正规广义Schmidt子组的组都是全幂的。