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The Tree consists of the Root G(2) ≅ (3,3), the infinite Trunk G(3) = G(3,3),G(4,5),G(5,7),…,G(m,2m-3),… , up to six infinite Branches starting with each node on the Trunk, finite Twigs, and Leaves (terminal nodes). We restrict the Tree to (isomorphism classes of) groups G in CBF(m,n) of order 3n and class m-1 that occur as automorphism groups Gal(K2|K) for quadratic base fields K with discriminant between -106 and 107. K2 denotes the 2nd Hilbert 3-class field of K. The Leaves (and certain other nodes1)) are represented together with their multiplet(s) of capitulation types distinguishing Ground States and Excited States. (These concepts are dependent on the kind of base field!) The finite Twigs with the Leaves of capitulation types D.5, D.10, and the Ground States of G.19, H.4 are represented in a position above the groups of lower than second maximal class only to save space. Their nodes are groups G of second maximal class and G´ is of rank 3, with the exception of G.19. 1) Since January 1st 2008 we know that an internal node can occur as automorphism group Gal(K2|K), as the capitulation type c.21 shows. |
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Annihilator Modules Let G be a 2-stage metabelian 3-group with two generators S1,S2. Denote by A := [S2,S1] = S2-1S1-1S2S1 the main commutator of G´. The operation of G on the abelian normal subgroup G´ by conjugation AS := S-1AS can be extended to an operation of the group ring Z[S1,S2] on G´. The mapping h: (Z[S1,S2],+)-->(G,*), f(S1,S2)-->Af(S1,S2) is a group homomorphism with image im(h) = G´ (according to Furtwängler) and with kernel M := ker(h) = { f(S1,S2) in Z[S1,S2] | Af(S1,S2) = 1 }, the symbolic order (or annihilator module) of A. Consequently, by the homomorphism theorem, we have an isomorphism (Z[S1,S2]/M,+)-->(G´,*). The symbolic orders M which actually occur for the main commutator A of 2-stage metabelian 3-groups G with commutator factor group of type (3,3) are generated by powers of the invertible linear polynomials X := S1-1 and Y := S2-1. Hence, we replace the group ring Z[S1,S2] by the isomorphic ring Z[X,Y] and we display the inclusion relations among the various symbolic orders M in a network diagram, using the following notation: L := (3,X,Y), L2 := (3,X2,XY,Y2), Ya := (Xa,Y,3+3X+X2), Xa := (Xa,XY,Y2,3+3X+X2), Ra,b := (Xa,XY,Yb,3+3X+3Y+X2+Y2). |
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Structure of the Commutator Subgroup Using the isomorphism (Z[S1,S2]/M,+)-->(G´,*), the structure of the commutator subgroup G´ of G can also be determined by means of ring theory. For the various symbolic orders M, we obtain: Z[X,Y]/L = <1> of type (3) (cyclic), Z[X,Y]/Ya = <1,X> of type (3u+1,3u), if a = 2u+1 is odd, and of type (3u,3u), if a = 2u is even (Rank 2), Z[X,Y]/L2 = <1,X,Y> of type (3,3,3) (Rank 3), Z[X,Y]/Xa = <1,X,Y> of type (3u+1,3u,3), if a = 2u+1 is odd, and of type (3u,3u,3), if a = 2u is even (Rank 3), Z[X,Y]/Ra,b = <1,X,Y,Y2> of type (3u,3u,3v,3v-1), if a >= b > 2, a = 2u, b = 2v are even (Rank 4). |
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Capitulation Types Now we give the connection of the symbolic orders with the capitulation types: First the symbolic orders of the main commutator of groups in CBFa(m,n): Z[X,Y] is associated with a degenerate form of type a.1, L with the unique type A.1, L2 with the types in section D (D.5, D.10), Y2 with the ground state of types a.2, a.3* and a.3, Y4 with the first excited state of types a.2 and a.3, X3 with the ground state of types in section c (c.18, c.21), X5 with the first excited state of types in section c, X4 with the ground state of types in section E (E.6, E.8, E.9, E.14), X6 with the first excited state of types in section E, R4,3 with the ground state of types in section d (d.19, d.23, d.25), R4,4 with the ground state of types in section F (F.7, F.11, F.12, F.13), R6,4 with the first excited state of types in section F. Second the symbolic orders of the main commutator of groups in CBFb(m,n): Y'4 is associated with the ground state of type a.1, Z5 with the ground state of type G.16, V'5,5 with the regular excited state of type G.16, V5,5 with the irregular excited state of type G.16, Z with the ground state of type G.19, T'5,5 with the regular excited state of type G.19, T5,5 with the irregular excited state of type G.19, V4,4 with the ground state of type b.10, Z' with the ground state of type H.4, Z'5 with the first excited state of type H.4, Z'7 with the second excited state of type H.4, V'5,5 with the regular excited state of type H.4, V5,5 with the irregular excited state of type H.4. |
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