3-groups with transfer kernel type f

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DANIEL C. MAYER. Abstract. For certain metabelian 3-groups G with abelianization G/G of type (3, 3) and transfer kernel type F, the cover cov(G) is determined, ...
3-GROUPS WITH TRANSFER KERNEL TYPE F DANIEL C. MAYER Abstract. For certain metabelian 3-groups G with abelianization G/G0 of type (3, 3) and transfer kernel type F, the cover cov(G) is determined, that is the set of all non-metabelian 3-groups H whose second derived quotient H/H 00 is isomorphic to G.

1. Introduction In several recent presentations and papers [17, 18, 19, 5, 20, 21, 22], we succeeded in determining the cover cov(G) of all metabelian 3-groups G with transfer kernel type (TKT) E [14]. These groups share the common coclass cc(G) = 2, the common class-1 quotient (abelianization) G/γ2 (G) ' h9, 2i and the common class-2 quotient G/γ3 (G) ' h27, 3i, in the notation of the SmallGroups library [1, 2]. Their class-3 quotient G/γ4 (G) is given by either h243, 6i for TKT E.6, κ(G) ∼ (1122), and E.14, κ(G) ∼ (3122), or h243, 8i for TKT E.8, κ(G) ∼ (2234), and E.9, κ(G) ∼ (2334) [12, 14]. In the present paper, we determine the cover cov(G) of certain metabelian 3-groups G with TKT F [14]. These groups can have any coclass cc(G) = r ≥ 3, but they share the common class-3 quotient G/γ4 (G) ' h243, 3i. Furthermore, they either arise as sporadic vertices of coclass graphs G(3, r) outside of coclass trees or as members of periodic infinite sequences on coclass trees Tr ⊂ G(3, r) [6, 7]. 2. Sporadic vertices outside of coclass trees With respect to number theoretic applications, we focus on 3-groups G of odd class cl(G) = c ≥ 5 and even coclass cc(G) = r ≥ 4, which admit an automorphism σ ∈ Aut(G) acting as inversion σ : x 7→ x−1 on the abelianization G/G0 . Such groups are called σ-groups. We begin with groups of minimal order forming the ground state of TKT F. Theorem 2.1. There exist 13 metabelian 3-groups G of order |G| = 39 , class cl(G) = 5, and coclass cc(G) = 4, having transfer kernel types (TKTs) in section F. They are immediate descendants of depth 2 of the parent group P = h2187, 64i in the SmallGroups library [1, 2], that is, their last lower central γ5 (G) is of type (3, 3) and P ' G/γ5 (G) is their common class-4 quotient. In the notation of the ANUPQ package [9] of GAP [8] and MAGMA [11], they are given by G = P −#2; j with  j ∈ {36, 38} for TKT F.11, κ(G) = (2133),    j ∈ {41, 47, 50, 52} for TKT F.13, κ(G) = (2123),  j ∈ {43, 46, 51, 53} for TKT F.12, κ(G) = (2124),    j ∈ {55, 56, 58} for TKT F.7, κ(G) = (2121). Proof. We use the p-group generation algorithm [24, 25, 10] as implemented in the computational algebra system Magma [3, 4, 11] to construct these 13 groups. We start with P :=SmallGroup(2187, 64), c :=NilpotencyClass(P ), call the Magma function D :=descendants(P, c+1 :step sizes:= [2]), and test all members of the list D for a suitable TKT in section F, making use of our own implementation of the Artin transfer homomorphisms.  Date: April 20, 2015. 2000 Mathematics Subject Classification. Primary 20D15, 20F12, 20F14. Key words and phrases. power-commutator presentations, 3-groups of derived lengths 2 and 3, transfer kernel type, cover, balanced cover, central series, lattice of normal subgroups, pruned coclass trees. Research supported by the Austrian Science Fund (FWF): P 26008-N25. 1

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The following groups of bigger order form the first excited state of TKT F. Theorem 2.2. There exist 13 metabelian 3-groups G of order |G| = 313 , class cl(G) = 7, and coclass cc(G) = 6, having transfer kernel types (TKTs) in section F. They are immediate descendants of depth 2 of the parent group P = h2187, 64i − #2; 33 − #2; 25 in the notation of the ANUPQ package [9] of GAP [8] and MAGMA [11], that is, their last lower central γ7 (G) is of type (3, 3) and P ' G/γ7 (G) is their common class-6 quotient. They are given by G = P − #2; j with  j ∈ {40, 42} for TKT F.11, κ(G) = (2133),    j ∈ {45, 51, 54, 56} for TKT F.13, κ(G) = (2123),  j ∈ {47, 50, 55, 57} for TKT F.12, κ(G) = (2124),    j ∈ {59, 60, 62} for TKT F.7, κ(G) = (2121). Remark 2.1. The group h2187, 64i − #2; 33 is a sibling of the 13 groups in Theorem 2.1 and the grandparent of the 13 groups in Theorem 2.2. Proof. Again, we use the p-group generation algorithm [24, 25, 10] as implemented in the computational algebra system Magma [3, 4, 11] to construct these 13 groups. We start with P = h2187, 64i− #2; 33−#2; 25, given by its compact presentation s, i.e. P :=PCGroup(s), c :=NilpotencyClass(P ), call the Magma function D :=descendants(P, c + 1 :step sizes:= [2]), and test all members of the list D for a suitable TKT in section F, making use of our own implementation of the Artin transfer homomorphisms.  The following Figure 1 shows the complete normal lattice of the groups G in Theorems 2.2 and 2.1. The lattice consists of diamonds of type (3, 3). Omitting two of the four cyclic subgroups, we draw each diamond as a square standing on one of its vertices. The members γj (G), 1 ≤ j ≤ cl(G) + 1, of the lower central series are indicated by tiny full discs. Except for the mandatory bottleneck γ2 (G)/γ3 (G), all factors γj (G)/γj+1 (G) are bicyclic. Thus we call G a BF-group (as opposed to a CF-group [23]). For such groups, the upper central series ζj (G), 0 ≤ j ≤ cl(G), is just the reverse lower central series. To enable a comparison, we emphasize that the smallest Schur σ-groups of order 35 with TKT D, i.e. the two groups h243, 5|7i, have a similar but simpler normal structure [13, 15, 16]. Figure 1. 3-groups of orders 313 , 39 with TKT F, and of order 35 with TKT D. order 3n r = 6, c = 7 6 r 1594323 313 @ @ 531441 312 @ @ r 11 177147 3 r 59049 310 r = 4, c = 5 @ @ r 9 19683 3 @ @ @ @ @r @ 6561 38 @ @ @ @ @ @ @ @ r 7 2187 3 @ @r @ @ @ @ @ @ r 729 36 r = 2, c = 3 @ @ @ @ @ @ @ @ @ @ @ @ r 5 243 3 @ @ @ @ @ @ @ @ @ @ @r @ @ @r @ @ 81 34 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ r 27 33 @ @ @ @ @r @ @ @ @ @r @ @ r 9 32 @ @ @ @ @ @ @ @ @ @ 3 @ @ @ @ @ @ r r r 1

3-GROUPS WITH TRANSFER KERNEL TYPE F

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3. Construction of the cover Theorem 3.1. The members H of minimal order |H| = 310 , class cl(H) = 5, coclass cc(H) = 5, and derived length dl(H) = 3, of the cover cov(G) = {H | dl(H) ≥ 3, H/H 00 ' G} of the 13 ground state groups G = P − #2; j with TKT F in Theorem 2.1 are immediate descendants of depth 3 of the parent group P = h2187, 64i, that is, their last lower central γ5 (H) is of type (3, 3, 3) and P ' H/γ5 (H) is their common class-4 quotient. They are of the form H = P − #3; ` with identifiers ` given in the following table, where terminal and capable groups are distinguished. Table 1. Cover groups of order 310 of 3-groups of order 39 with TKT F

j 36 38 41 47 50 52 43 46 51 53 55 56 58

terminal for ` = 140, 141 143, 144 148, 149 158, 159 162, 171 164, 166 151, 152 156, 157 163, 176 165, 177 168, 178 169 172

capable total for ` = count 239, 254, 260, 310, 313, 316 8 240, 255, 261, 268, 271, 274 8 281, 296, 302, 312, 315, 318 8 269, 272, 275, 324, 339, 345 8 242, 248, 263, 325, 331, 346 8 243, 249, 264, 283, 289, 304 8 270, 273, 276, 282, 297, 303 8 311, 314, 317, 323, 338, 344 8 245, 251, 257, 328, 334, 340 8 246, 252, 258, 286, 292, 298 8 287, 293, 299, 330, 336, 342 8 285, 291, 306 4 326, 332, 347 4

Proof. We use the p-group generation algorithm [24, 25, 10] as implemented in the computational algebra system Magma [3, 4, 11] to construct these 96 groups. We start with P :=SmallGroup(2187, 64), c :=NilpotencyClass(P ), call the Magma function D :=descendants(P, c+1 :step sizes:= [3]), and test all members of the list D for a suitable TKT in section F, making use of our own implementation of the Artin transfer homomorphisms. Finally we check the 96 second derived quotients H/H 00 against the 13 groups G of Theorem 2.1 for isomorphism H/H 00 ' G, stopping at the first isomorphism encountered. The two groups G with identifiers 56, 58 of TKT F.7 turn out to be exceptional. 

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References [1] H.U. Besche, B. Eick, and E.A. O’Brien, A millennium project: constructing small groups, Int. J. Algebra Comput. 12 (2002), 623-644. [2] H.U. Besche, B. Eick, and E.A. O’Brien, The SmallGroups Library — a Library of Groups of Small Order, 2005, an accepted and refereed GAP 4 package, available also in MAGMA. [3] W. Bosma, J. Cannon, and C. Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), 235–265. [4] W. Bosma, J.J. Cannon, C. Fieker, and A. Steels (eds.), Handbook of Magma functions (Edition 2.21, Sydney, 2015). [5] M.R. Bush and D.C. Mayer, 3-class field towers of exact length 3, J. Number Theory 147 (2015), 766–777, DOI 10.1016/j.jnt.2014.08.010. ´ [6] M. du Sautoy, Counting p-groups and nilpotent groups, Inst. Hautes Etudes Sci. Publ. Math. 92 (2000) 63–112. [7] B. Eick and C. Leedham-Green, On the classification of prime-power groups by coclass, Bull. London Math. Soc. 40 (2) (2008), 274–288. [8] The GAP Group, GAP – Groups, Algorithms, and Programming — a System for Computational Discrete Algebra, Version 4.7.7, Aachen, Braunschweig, Fort Collins, St. Andrews, 2015, (http://www.gap-system.org). [9] G. Gamble, W. Nickel, and E.A. O’Brien, ANU p-Quotient — p-Quotient and p-Group Generation Algorithms, 2006, an accepted GAP 4 package, available also in MAGMA. [10] D.F. Holt, B. Eick, and E.A. O’Brien, Handbook of computational group theory, Discrete mathematics and its applications, Chapman and Hall/CRC Press, Boca Raton, 2005. [11] The MAGMA Group, MAGMA Computational Algebra System, Version 2.21-3, Sydney, 2015, (http://magma.maths.usyd.edu.au). [12] D.C. Mayer, Principalization in complex S3 -fields, Congressus Numerantium 80 (1991), 73–87. (Proceedings of the Twentieth Manitoba Conference on Numerical Mathematics and Computing, Univ. of Manitoba, Winnipeg, Canada, 1990.) [13] D.C. Mayer, The second p-class group of a number field, Int. J. Number Theory 8 (2012), no. 2, 471–505, DOI 10.1142/S179304211250025X. [14] D.C. Mayer, Transfers of metabelian p-groups, Monatsh. Math. 166 (2012), no. 3–4, 467–495, DOI 10.1007/s00605-010-0277-x. [15] D.C. Mayer, Principalization algorithm via class group structure, J. Th´ eor. Nombres Bordeaux 26 (2014), no. 2, 415–464. [16] D.C. Mayer, The distribution of second p-class groups on coclass graphs, J. Th´ eor. Nombres Bordeaux 25 (2013) no. 2, 401–456, DOI 10.5802/jtnb842. (27th Journ´ ees Arithm´ etiques, Faculty of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania, 2011). [17] D.C. Mayer and M.F. Newman, Finite 3-groups as viewed from class field theory, Groups St Andrews 2013, Univ. of St Andrews, Fife, Scotland. ¨ [18] D.C. Mayer, M.R. Bush, and M.F. Newman, 3-class field towers of exact length 3, 18th OMG Congress and 123rd Annual DMV Meeting 2013, Univ. of Innsbruck, Tyrol, Austria. [19] D.C. Mayer, M.R. Bush, and M.F. Newman, Class towers and capitulation over quadratic fields, West Coast Number Theory 2013, Asilomar Conference Center, Pacific Grove, Monterey, California, USA. [20] D.C. Mayer, Periodic bifurcations in descendant trees of finite p-groups, Adv. Pure Math., 5 (2015) no. 4, 162–195, DOI 10.4236/apm.2015.54020, Special Issue on Group Theory, March 2015. [21] D.C. Mayer, Index-p abelianization data of p-class tower groups, Adv. Pure Math., 5 (2015) no. 5, 286–313, DOI 10.4236/apm.2015.55029, Special Issue on Number Theory and Cryptography, April 2015. [22] D.C. Mayer, Periodic sequences of p-class tower groups, to appear in J. Appl. Math. Phys., International Conference on Groups and Algebras, Shanghai, July 2015. [23] B. Nebelung, Klassifikation metabelscher 3-Gruppen mit Faktorkommutatorgruppe vom Typ (3, 3) und Anwendung auf das Kapitulationsproblem (Inauguraldissertation, Universit¨ at zu K¨ oln, 1989). [24] M.F. Newman, Determination of groups of prime-power order, pp. 73–84, in: Group Theory, Canberra, 1975, Lecture Notes in Math., vol. 573, Springer, Berlin, 1977. [25] E.A. O’Brien, The p-group generation algorithm, J. Symbolic Comput. 9 (1990), 677–698. Naglergasse 53, 8010 Graz, Austria E-mail address: [email protected] URL: http://www.algebra.at