Introduction Preliminaries Containers Cardinality of C ( A , B ) The Main Theorem On Commutation Semigroups of Dihedral Groups Darien DeWolf, Charles Edmunds, Christopher Levy October 13, 2012
Introduction Preliminaries Containers Cardinality of C ( A , B ) The Main Theorem Some Definitions to Start Definition For any group G with g ∈ G , the right and left commutation mappings associated with g are the mappings ρ ( g ) and λ ( g ) from G to G defined as ( x ) ρ ( g ) = [ x , g ] ( x ) λ ( g ) = [ g , x ] , where the commutator of g and h is defined as [ g , h ] = g − 1 h − 1 gh .
Introduction Preliminaries Containers Cardinality of C ( A , B ) The Main Theorem Definition The set M ( G ) of all mappings from G to G forms a semigroup under composition of mappings. Definition The right commutation semigroup of G , P ( G ), is the subsemigroup of M ( G ) generated by the set of ρ -maps and the left commutation semigroup of G , Λ( G ), is the subsemigroup of M ( G ) generated by the set of λ -maps .
Introduction Preliminaries Containers Cardinality of C ( A , B ) The Main Theorem Definition The dihedral group of order 2 m has presentation � a , b ; a m = 1 , b 2 = 1 , a b = a − 1 � D m = , where the conjugate of a by b is denoted a b = b − 1 ab .
Introduction Preliminaries Containers Cardinality of C ( A , B ) The Main Theorem An Anomaly Presents Itself Though P ( G ) and Λ( G ) have apparently symmetric definitions, it is not true in general that P ( G ) = Λ( G ) . In fact it is not even true in general that P ( G ) ∼ = Λ( G ) , for | P ( D 3 ) | = 6 � = 9 = | Λ( D 3 ) | . We then ask ourselves: “What conditions on the group can impose to force its commutation semigroups to be of equal order, isomorphic, equal?” We will focus now on dihedral groups.
Introduction Preliminaries Containers Cardinality of C ( A , B ) The Main Theorem Some Preliminaries Notation For each s ≥ 0 let α s = ( − 1) s and β s = ( − 1) s − 1. Since the values of α s and β s are unique up to parity, it will cause no ambiguity to view s as an element of Z 2 . Lemma Let D m be the dihedral group with presentation as above. For each i , r ∈ Z m and j , s ∈ Z 2 : ( a i b j ) ρ ( a r b s ) = a N ρ and ( a i b j ) λ ( a r b s ) = a N λ , where N ρ ≡ i α j β s − r α s β j ≡ ( − 2) α js ( is − jr )(mod m ) and N λ ≡ − N ρ ≡ 2 α js ( is − jr )(mod m ) .
Introduction Preliminaries Containers Cardinality of C ( A , B ) The Main Theorem Definition For each pair ( A , B ) ∈ Z m × Z m we define a µ -map µ ( A , B ) : D m → D m by ( a i b j ) µ ( A , B ) = a N µ , where N µ = Ai α j − B β j .
Introduction Preliminaries Containers Cardinality of C ( A , B ) The Main Theorem Lemma For each r ∈ Z m and s ∈ Z 2 , (i) ρ ( a r b s ) = µ ( β s , r α s ) , (ii) λ ( a r b s ) = µ ( − β s , − r α s ) . Lemma For each A , A ′ ∈ Z m and B , B ′ ∈ Z 2 , µ ( A , B ) ◦ µ ( A ′ , B ′ ) = µ ( AA ′ , BA ′ ) .
Introduction Preliminaries Containers Cardinality of C ( A , B ) The Main Theorem Containers of µ -maps Definition If A , B ∈ Z m , the ( A , B ) -container is defined as C ( A , B ) = { µ ( A , xB ) : x ∈ Z m } .
Introduction Preliminaries Containers Cardinality of C ( A , B ) The Main Theorem Lemma For all A , A ′ , B , B ′ ∈ Z m , C ( A , B ) ∩ C ( A ′ , B ′ ) � = ∅ if and only if A ≡ A ′ (mod m ) . Lemma P ( D n ) ⊇ { ρ g } = C (0 , 1) ˙ ∪ C ( − 2 , 1) and Λ( D n ) ⊇ { λ g } = C (0 , 1) ˙ ∪ C (2 , 1) .
Introduction Preliminaries Containers Cardinality of C ( A , B ) The Main Theorem Definition For any two containers C ( A , B ) and C ( A ′ , B ′ ), we define their product as: � � C ( A , B ) ◦ C ( A ′ , B ′ ) = µ 1 ◦ µ 2 : µ 1 ∈ C ( A , B ) , µ 2 ∈ C ( A ′ , B ′ ) .
Introduction Preliminaries Containers Cardinality of C ( A , B ) The Main Theorem Lemma For A , A ′ ∈ Z m and B , B ′ ∈ Z 2 , C ( A , B ) ◦ C ( A ′ , B ′ ) = C ( AA ′ , BA ′ ) . Lemma For A , B ∈ Z m , (i) C (0 , 1) ◦ C ( A , B ) ⊆ C (0 , 1) , (ii) C ( A , B ) ◦ C (0 , 1) ⊆ C (0 , 1) .
Introduction Preliminaries Containers Cardinality of C ( A , B ) The Main Theorem Definition For x ∈ Z , let ind m ( x ) be the smallest positive integer such that there is a per m ( x ) ∈ Z + with x ind m ( x ) = x ind m ( x )+ per m ( x ) and per m ( x ) the least such positive integer. Here ind m ( x ) and per m ( x ) are called the index and the period of x , Lemma If m = 2 ℓ n ≥ 3 with n odd, ℓ ≥ 0 , and n ≥ 1 , then for x ∈ {− 2 , 2 } (i) if m is odd, then ind m ( x ) = 1 and per m ( x ) = ord m ( x ) , (ii) if m is even and n > 1 , then ind m ( x ) = ℓ , (iii) if m is even and n = 1 , then ind m ( x ) = ℓ and per m ( x ) = 1 .
Introduction Preliminaries Containers Cardinality of C ( A , B ) The Main Theorem Theorem For m = 2 ℓ n ≥ 3 with n odd, ℓ ≥ 0 , and n ≥ 1 , � t � � C (( − 2) i , ( − 2) i − 1 ) P ( D m ) = C (0 , 1) ∪ , i =1 (i) ord m ( − 2) for ℓ = 0 , n > 1 where t = ℓ + per m ( − 2) − 1 for ℓ > 0 , n > 1 ℓ − 1 for ℓ > 0 , n = 1 � t ′ � � C (2 i , 2 i − 1 ) Λ( D m ) = C (0 , 1) ∪ , i =1 (ii) ord m (2) for ℓ = 0 , n > 1 where t ′ = ℓ + per m (2) − 1 for ℓ > 0 , n > 1 , ℓ − 1 for ℓ > 0 , n = 1 and these unions are disjoint.
Introduction Preliminaries Containers Cardinality of C ( A , B ) The Main Theorem Definition The upper central series of a group G is the series of subgroups of G , Z 0 ( G ) ≤ Z 1 ( G ) ≤ · · · ≤ Z n ( G ) ≤ · · · with Z 0 ( G ) = { 1 } and Z n ( G ) = { g ∈ G : [ g , g 1 , g 2 , . . . , g n ] = 1 , for all g 1 , g 2 , . . . , g n ∈ G } . We call Z n ( G ) the n -th-centre of G and, where no ambiguity arises, denote it Z n .
Introduction Preliminaries Containers Cardinality of C ( A , B ) The Main Theorem Theorem (a) If u ≥ 0 and m is odd, then Z u ( D m ) = { 1 } , (b) If u ≥ 0 and m is even with m = 2 ℓ n ( n > 0 and n odd ) , then (i) if n > 1 , then a N : N = (2 ℓ − u n ) x and 0 ≤ x < 2 u � � � , u < ℓ Z u ( D m ) = a nx : 0 ≤ x < 2 ℓ � � , u ≥ ℓ (ii) if n = 1 , then a N : N = 2 ℓ − u x and 0 ≤ x < 2 u � � � , u < ℓ Z u ( D m ) = D m , u ≥ ℓ
Introduction Preliminaries Containers Cardinality of C ( A , B ) The Main Theorem Theorem If u > 0 and Z u ( D m ) ≤ � a � , then m � � � C (( − 2) u , ( − 2) u − 1 ) (i) � = | Z u ( D m ) | , � � m � = � � C (2 u , 2 u − 1 ) � (ii) | Z u ( D m ) | . Lemma |C (0 , 1) | = m | Z 1 | .
Introduction Preliminaries Containers Cardinality of C ( A , B ) The Main Theorem The Main Theorem Theorem If m = 2 ℓ n > 3 with n odd, t − 1 � � 1 1 � | P ( D m ) | = m | Z 1 | + | Z i | i =1 (i) 1 + ord m ( − 2) for ℓ = 0 , n > 1 where t = ℓ + per m ( − 2) for ℓ > 0 , n > 1 ℓ for ℓ > 0 , n = 1 t ′ − 1 � � 1 1 � | Λ( D m ) | = m | Z 1 | + | Z i | i =1 (ii) 1 + ord m (2) for ℓ = 0 , n > 1 where t ′ = ℓ + per m (2) for ℓ > 0 , n > 1 ℓ for ℓ > 0 , n = 1
Recommend
More recommend
