New progress on factorized groups and subgroup permutability Paz Arroyo-Jordá Instituto Universitario de Matemática Pura y Aplicada Universidad Politécnica de Valencia, Spain Groups St Andrews 2013 in St Andrews St Andrews, 3rd-11th August 2013 in collaboration with M. Arroyo-Jordá, A. Martínez-Pastor and M.D. Pérez-Ramos P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 1 / 30
Introduction General problem Factorized groups All groups considered will be finite Factorized groups: A and B subgroups of a group G G = AB How the structure of the factors A and B affects the structure of the whole group G ? How the structure of the group G affects the structure of A and B ? P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 2 / 30
Introduction General problem Factorized groups Natural approach: Classes of groups A class of groups is a collection F of groups with the property that if G ∈ F and G ∼ = H , then H ∈ F P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 3 / 30
Introduction General problem Factorized groups Natural approach: Classes of groups A class of groups is a collection F of groups with the property that if G ∈ F and G ∼ = H , then H ∈ F Question Let F be a class of groups and G = AB a factorized group: A , B ∈ F = ⇒ G ∈ F ? G ∈ F = ⇒ A , B ∈ F ? P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 3 / 30
Introduction Formations Definitions A formation is a class F of groups with the following properties: Every homomorphic image of an F -group is an F -group. If G / M and G / N ∈ F , then G / ( M ∩ N ) ∈ F F a formation: the F -residual G F of G is the smallest normal subgroup of G such that G / G F ∈ F The formation F is said to be saturated if G / Φ( G ) ∈ F , then G ∈ F . P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 4 / 30
Introduction Products of supersoluble groups Starting point G = AB : A , B ∈ U , A , B � G � = ⇒ G ∈ U Example Q = � x , y � ∼ V = � a , b � ∼ = Q 8 , = C 5 × C 5 G = [ V ] Q the semidirect product of V with Q G = AB with A = V � x � and B = V � y � A , B ∈ U , A , B � G , G �∈ U P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 5 / 30
Introduction Products of supersoluble groups Starting point G = AB : A , B ∈ U , A , B � G � = ⇒ G ∈ U Example Q = � x , y � ∼ V = � a , b � ∼ = Q 8 , = C 5 × C 5 G = [ V ] Q the semidirect product of V with Q G = AB with A = V � x � and B = V � y � A , B ∈ U , A , B � G , G �∈ U A , B ∈ U , A , B � G + additional conditions ⇒ G ∈ U G = AB : = (Baer, 57) G ′ ∈ N (Friesen,71) ( | G : A | , | G : B | ) = 1 P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 5 / 30
Introduction Products of supersoluble groups Permutability properties If G = AB is a central product of the subgroups A and B , then: A , B ∈ U = ⇒ G ∈ U More generally, if F is any formation: A , B ∈ F = ⇒ G ∈ F (In particular, this holds when G = A × B is a direct product.) P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 6 / 30
Introduction Products of supersoluble groups Permutability properties If G = AB is a central product of the subgroups A and B , then: A , B ∈ U = ⇒ G ∈ U More generally, if F is any formation: A , B ∈ F = ⇒ G ∈ F (In particular, this holds when G = A × B is a direct product.) Let G = AB a factorized group: A , B ∈ U ( or F )+ permutability properties = ⇒ G ∈ U ( or F ) P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 6 / 30
Permutability properties Total permutability Total permutability Definition Let G be a group and let A and B be subgroups of G . It is said that A and B are totally permutable if every subgroup of A permutes with every subgroup of B . Theorem (Asaad,Shaalan, 89) If G = AB is the product of the totally permutable subgroups A and B, then A , B ∈ U = ⇒ G ∈ U P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 7 / 30
Permutability properties Total permutability Total permutability and formations (Maier,92; Carocca,96;Ballester-Bolinches, Pedraza-Aguilera, Pérez-Ramos, 96-98) Let F be a formation such that U ⊆ F . Let the group G = G 1 G 2 · · · G r be a product of pairwise totally permutable subgroups G 1 , G 2 , . . . , G r , r ≥ 2. Then: Theorem If G i ∈ F ∀ i ∈ { 1 , . . . , r } , then G ∈ F . Assume in addition that F is either saturated or F ⊆ S . If G ∈ F , then G i ∈ F , ∀ i ∈ { 1 , . . . , r } . Corollary If F is either saturated or F ⊆ S , then: G F = G F 1 G F 2 . . . G F r . P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 8 / 30
Permutability properties Conditional permutability Conditional permutability Definitions (Qian,Zhu,98) (Guo, Shum, Skiba, 05) Let G be a group and let A and B be subgroups of G . A and B are conditionally permutable in G (c-permutable), if AB g = B g A for some g ∈ G . A and B are totally c-permutable if every subgroup of A is c-permutable in G with every subgroup of B . Permutable = ⇒ = Conditionally Permutable �⇐ Example Let X and Y be two 2-Sylow subgroups of S 3 . Then X permutes with Y g for some g ∈ S 3 , but X does not permute with Y . P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 9 / 30
Permutability properties Conditional permutability Total c-permutability and supersolubility Theorem (Arroyo-Jordá, AJ, Martínez-Pastor,Pérez-Ramos,10) Let G = AB be the product of the totally c-permutable subgroups A and B. Then: G U = A U B U P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 10 / 30
Permutability properties Conditional permutability Total c-permutability and supersolubility Theorem (Arroyo-Jordá, AJ, Martínez-Pastor,Pérez-Ramos,10) Let G = AB be the product of the totally c-permutable subgroups A and B. Then: G U = A U B U In particular, A , B ∈ U ⇐ ⇒ G ∈ U P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 10 / 30
Permutability properties Conditional permutability Total c-permutability and supersolubility Theorem (Arroyo-Jordá, AJ, Martínez-Pastor,Pérez-Ramos,10) Let G = AB be the product of the totally c-permutable subgroups A and B. Then: G U = A U B U In particular, A , B ∈ U ⇐ ⇒ G ∈ U Corollary , PR, 10) Let G = AB be the product of the totally (AJ, AJ, MP c-permutable subgroups A and B and let p be a prime. If A , B are p-supersoluble, then G is p-supersoluble. P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 10 / 30
Permutability properties Conditional permutability Total c-permutability and saturated formations Question Are saturated formations F (of soluble groups) containing U closed under taking products of totally c-permutable subgroups? Example Take G = S 4 = AB , A = A 4 and B ∼ = C 2 generated by a transposition. Then A and B are totally c-permutable in G . Let F = N 2 , the saturated formation of metanilpotent groups. Notice U ⊆ N 2 ⊆ S . Then: A , B ∈ F but G �∈ F . In particular, G F � = A F B F . P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 11 / 30
Permutability properties Conditional permutability Conditional permutability Remark c-permutability fails to satisfy the property of persistence in intermediate subgroups. Example Let G = S 4 and let Y ∼ = C 2 generated by a transposition. Let V be the normal subgroup of G of order 4 and X a subgroup of V of order 2, X � = Z ( VY ) . Then X and Y are c-permutable in G X and Y are not c-permutable in � X , Y � . P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 12 / 30
Permutability properties Complete c-permutability Complete c-permutability Definitions (Guo, Shum, Skiba, 05) Let G be a group and let A and B be subgroups of G . A and B are completely c-permutable in G (cc-permutable), if AB g = B g A for some g ∈ � A , B � . A and B are totally completely c-permutable (tcc-permutable) if every subgroup of A is completely c-permutable in G with every subgroup of B . ⇒ ⇒ = = Totally permutable Totally completely c-permutable Totally c-permutable �⇐ �⇐ = = P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 13 / 30
Permutability properties Complete c-permutability Complete c-permutability and supersolubility G = AB , A , B totally c-permutable, G U = A U B U Corollary (Guo, Shum, Skiba, 06) Let G = AB be a product of the tcc-permutable subgroups A and B. If A , B ∈ U , then G ∈ U . P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 14 / 30
Permutability properties Complete c-permutability Complete c-permutability and supersolubility G = AB , A , B totally c-permutable, G U = A U B U Corollary (Guo, Shum, Skiba, 06) Let G = AB be a product of the tcc-permutable subgroups A and B. If A , B ∈ U , then G ∈ U . Let G = AB be the product of the tcc-permutable subgroups A and B and let p be a prime. If A , B are p-supersoluble, then G is p-supersoluble. P . Arroyo-Jordá (IUMPA-UPV) Factorized groups St Andrews 2013 14 / 30
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