On the capabillity of p -groups of class two and prime exponent - PowerPoint PPT Presentation

On the capabillity of p -groups of class two and prime exponent Arturo Magidin University of Louisiana at Lafayette Groups St Andrews 2013 Arturo Magidin

Capability Definition A group G is capable if and only if there exists K such that G ∼ = K / Z ( K ) . Arturo Magidin

The gold standard theorem Theorem (Baer) Let G be a finitely generated abelian group, and write G = C a 1 ⊕ · · · ⊕ C a r , with a 1 | a 2 | · · · | a r . Then G is capable if and only if r ≥ 2 and a r − 1 = a r . Arturo Magidin

p -groups of class 2 Specific p -group of class 2 can be checked for capability easily enough. Arturo Magidin

p -groups of class 2 Specific p -group of class 2 can be checked for capability easily enough. But no full characterization of capability for p -groups of class 2 (similar to Baer’s for abelian) exists, and current techniques do not seem sufficient. Arturo Magidin

p -groups of class 2 Specific p -group of class 2 can be checked for capability easily enough. But no full characterization of capability for p -groups of class 2 (similar to Baer’s for abelian) exists, and current techniques do not seem sufficient. For the subclass of p -groups of class 2 and exponent p , there is some hope: there are known sufficient and known necessary conditions for capability. Sufficient conditions are notoriously hard to come by. Arturo Magidin

Some known necessary conditions for groups of class 2 and exponent p If G is a nontrivial central product, G = AB with A ⊆ C ( B ) , B ⊆ C ( A ) , [ A , A ] ∩ [ B , B ] � = { 1 } , then G is not capable. Arturo Magidin

Some known necessary conditions for groups of class 2 and exponent p If G is a nontrivial central product, G = AB with A ⊆ C ( B ) , B ⊆ C ( A ) , [ A , A ] ∩ [ B , B ] � = { 1 } , then G is not capable. In particular, if G is extraspecial, then G is capable if and only if | G | = p 3 and G is of exponent p . (Beyl, Felgner, Schmidt; 1979). Arturo Magidin

Some known necessary conditions for groups of class 2 and exponent p If G is a nontrivial central product, G = AB with A ⊆ C ( B ) , B ⊆ C ( A ) , [ A , A ] ∩ [ B , B ] � = { 1 } , then G is not capable. In particular, if G is extraspecial, then G is capable if and only if | G | = p 3 and G is of exponent p . (Beyl, Felgner, Schmidt; 1979). If G is capable, and rank ([ G , G ]) = k , then rank ( G / Z ( G )) ≤ 2 k + � k � (Heineken and Nikolova; 1996). 2 Arturo Magidin

Some known necessary conditions for groups of class 2 and exponent p If G is a nontrivial central product, G = AB with A ⊆ C ( B ) , B ⊆ C ( A ) , [ A , A ] ∩ [ B , B ] � = { 1 } , then G is not capable. In particular, if G is extraspecial, then G is capable if and only if | G | = p 3 and G is of exponent p . (Beyl, Felgner, Schmidt; 1979). If G is capable, and rank ([ G , G ]) = k , then rank ( G / Z ( G )) ≤ 2 k + � k � (Heineken and Nikolova; 1996). 2 (Intuitively: if G is capable, then its commutator subgroup cannot be too small) Arturo Magidin

Some known necessary conditions for groups of class 2 and exponent p If g 1 , . . . , g n generated G , Z ( G ) = G ′ , and n � [ C G ( g i ) , C G ( g i )] � = 1 , i = 1 then G is not capable (follows easily from work of Ellis of 1996). Arturo Magidin

Some known necessary conditions for groups of class 2 and exponent p If g 1 , . . . , g n generated G , Z ( G ) = G ′ , and n � [ C G ( g i ) , C G ( g i )] � = 1 , i = 1 then G is not capable (follows easily from work of Ellis of 1996). Conjecture Let G be a p-group of exponent p with Z ( G ) = G ′ . Then G is capable if and only if for all minimal generating sets g 1 , . . . , g n N � � � C G ( g i ) , C G ( g i ) = 1 . i = 1 Arturo Magidin

Some known sufficient conditions for groups of class 2 and exponent p If g 1 , . . . , g n project onto a basis for G ab , and the nontrivial commutators among [ g j , g i ] are distinct and form a basis for [ G , G ] , then G is capable. (Ellis, 1996) Arturo Magidin

Some known sufficient conditions for groups of class 2 and exponent p If g 1 , . . . , g n project onto a basis for G ab , and the nontrivial commutators among [ g j , g i ] are distinct and form a basis for [ G , G ] , then G is capable. (Ellis, 1996) (Intuitively: if the relations among basic commutators are very simple, then G is capable) Arturo Magidin

Some known sufficient conditions for groups of class 2 and exponent p If g 1 , . . . , g n project onto a basis for G ab , and the nontrivial commutators among [ g j , g i ] are distinct and form a basis for [ G , G ] , then G is capable. (Ellis, 1996) (Intuitively: if the relations among basic commutators are very simple, then G is capable) If G is a coproduct, G = A ∐ N 2 B with A and B nontrivial, then G is capable (2008). Arturo Magidin

Some known sufficient conditions for groups of class 2 and exponent p If g 1 , . . . , g n project onto a basis for G ab , and the nontrivial commutators among [ g j , g i ] are distinct and form a basis for [ G , G ] , then G is capable. (Ellis, 1996) (Intuitively: if the relations among basic commutators are very simple, then G is capable) If G is a coproduct, G = A ∐ N 2 B with A and B nontrivial, then G is capable (2008). ( A ∐ N 2 B is the most general group of class 2 and exponent p that contains a copy of A and a copy of B ) Arturo Magidin

Other results If G is a p -group of class two and exponent p , then there exist groups G 1 and G 2 such that: G ≤ G 1 , G 2 ; Arturo Magidin

Other results If G is a p -group of class two and exponent p , then there exist groups G 1 and G 2 such that: G ≤ G 1 , G 2 ; G p 1 = G p 2 = { 1 } ; Arturo Magidin

Other results If G is a p -group of class two and exponent p , then there exist groups G 1 and G 2 such that: G ≤ G 1 , G 2 ; G p 1 = G p 2 = { 1 } ; Z ( G i ) = [ G i , G i ] , i = 1 , 2; Arturo Magidin

Other results If G is a p -group of class two and exponent p , then there exist groups G 1 and G 2 such that: G ≤ G 1 , G 2 ; G p 1 = G p 2 = { 1 } ; Z ( G i ) = [ G i , G i ] , i = 1 , 2; Neither is a nontrivial central product; Arturo Magidin

Other results If G is a p -group of class two and exponent p , then there exist groups G 1 and G 2 such that: G ≤ G 1 , G 2 ; G p 1 = G p 2 = { 1 } ; Z ( G i ) = [ G i , G i ] , i = 1 , 2; Neither is a nontrivial central product; G 1 is capable and G 2 is not capable. Arturo Magidin

Counterpart to Heineken-Nikolova I want to talk about a counterpart to: Theorem (Heineken, Nikolova) If G is capable, and rank ([ G , G ]) = k, then rank ( G / Z ( G )) ≤ 2 k + � k � . 2 Arturo Magidin

Counterpart to Heineken-Nikolova I want to talk about a counterpart to: Theorem (Heineken, Nikolova) If G is capable, and rank ([ G , G ]) = k, then rank ( G / Z ( G )) ≤ 2 k + � k � . 2 Namely, a result that says that if [ G , G ] is “sufficiently large”, then G is capable. Arturo Magidin

Set-up Let G be a group of class 2 and exponent p , Z ( G ) = [ G , G ] , and g 1 , . . . , g n projecting onto a basis for G ab . Arturo Magidin

Set-up Let G be a group of class 2 and exponent p , Z ( G ) = [ G , G ] , and g 1 , . . . , g n projecting onto a basis for G ab . Let F = � x 1 , . . . , x n � be the relatively free group of rank n , class 3, and exponent p . Arturo Magidin

Set-up Let G be a group of class 2 and exponent p , Z ( G ) = [ G , G ] , and g 1 , . . . , g n projecting onto a basis for G ab . Let F = � x 1 , . . . , x n � be the relatively free group of rank n , class 3, and exponent p . If ψ : F → G is induced by x i �→ g i , then ψ factors through F / F 3 , and we have: / Z ( F ) F − − − − − − − − − − − → F / F 3   / [ X , F ]   � / XF 3 � F / [ X , F ] − − − − − − − − − − − → G / ( XF 3 / [ X , F ]) Arturo Magidin

Example Say G = � g 1 , g 2 , g 3 | [ g 1 , g 2 ] = [ g 1 , g 3 ] , G 3 = 1 , G p = 1 � . Arturo Magidin

Example Say G = � g 1 , g 2 , g 3 | [ g 1 , g 2 ] = [ g 1 , g 3 ] , G 3 = 1 , G p = 1 � . We can take X = � [ x 1 , x 2 ][ x 1 , x 3 ] − 1 � ≤ F . Then F / XF 3 ∼ = G . Arturo Magidin

Example Say G = � g 1 , g 2 , g 3 | [ g 1 , g 2 ] = [ g 1 , g 3 ] , G 3 = 1 , G p = 1 � . We can take X = � [ x 1 , x 2 ][ x 1 , x 3 ] − 1 � ≤ F . Then F / XF 3 ∼ = G . In any witness to the capability of G , we would need to have � � [ x 1 , x 2 ][ x 1 , x 3 ] − 1 , x j = 1 , j = 1 , 2 , 3 ; i.e., [ X , F ] would have to be trivial. Arturo Magidin

Connection to epicenter Z ∗ ( G ) is the epicenter of G . It is the smallest normal subgroup of G for which G / N is capable. Arturo Magidin

Connection to epicenter Z ∗ ( G ) is the epicenter of G . It is the smallest normal subgroup of G for which G / N is capable. / Z ( F ) − − − − − − − − − − − → F / F 3 F   / [ X , F ]   � / XF 3 � F / [ X , F ] − − − − − − − − − − − → G / ( XF 3 / [ X , F ]) Arturo Magidin