Quantum mechanics

Permutable prime

Discrete mathematics

Fitting subgroup

Normal subgroup

Finite group

Subgroup

Physics

p-group

Mathematics

Partition (number theory)

Combinatorics

Symmetric group

Index of a subgroup

Group (periodic table)

Mathematica Slovaca

Let σ = { σ i ∣ i ∈ I } be some partition of the set of all primes ℙ, G be a finite group and σ ( G ) = { σ i ∣ σ i ∩ π ( G ) ≠ ∅}. G is said to be σ - primary if ∣ σ ( G )∣ ≤ 1. A subgroup H of G is said to be σ - subnormal in G if there exists a subgroup chain H = H 0 ≤ H 1 ≤ … ≤ H t = G such that either H i −1 is normal in H i or H i /( H i −1 ) H i is σ -primary for all i = 1, …, t . A set 𝓗 of subgroups of G is said to be a complete Hall σ - set of G if every non-identity member of 𝓗 is a Hall σ i -subgroup of G for some i and 𝓗 contains exactly one Hall σ i -subgroup of G for every σ i ∈ σ ( G ). Let 𝓗 be a complete Hall σ -set of G . A subgroup H of G is said to be 𝓗- permutable if HA = AH for all A ∈ 𝓗. We say that a subgroup H of G is weakly 𝓗- permutable in G if there exists a σ -subnormal subgroup T of G such that G = HT and H ∩ T ≤ H 𝓗 , where H 𝓗 is the subgroup of H generated by all those subgroups of H which are 𝓗-permutable. By using the weakly 𝓗-permutable subgroups, we establish some new criteria for a group G to be σ -soluble and supersoluble, and we also give the conditions under which a normal subgroup of G is hypercyclically embedded.

On weakly 𝓗-permutable subgroups of finite groups