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Journal articles on the topic 'Finite abelian group'

Author: Grafiati

Published: 4 June 2021

Last updated: 4 February 2022

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1

JAIN, VIVEK K., PRADEEP K. RAI, and MANOJ K. YADAV. "ON FINITE p-GROUPS WITH ABELIAN AUTOMORPHISM GROUP." International Journal of Algebra and Computation 23, no. 05 (August 2013): 1063–77. http://dx.doi.org/10.1142/s0218196713500161.

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We construct, for the first time, various types of specific non-special finite p-groups having abelian automorphism group. More specifically, we construct groups G with abelian automorphism group such that γ2(G) < Z(G) < Φ(G), where γ2(G), Z(G) and Φ(G) denote the commutator subgroup, the center and the Frattini subgroup of G respectively. For a finite p-group G with elementary abelian automorphism group, we show that at least one of the following two conditions holds true: (i) Z(G) = Φ(G) is elementary abelian; (ii) γ2(G) = Φ(G) is elementary abelian, where p is an odd prime. We construct examples to show the existence of groups G with elementary abelian automorphism group for which exactly one of the above two conditions holds true.

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2

Pranjali, Mukti Acharya, and Purnima Gupta. "Finite Abelian Group Labeling." Electronic Notes in Discrete Mathematics 48 (July 2015): 255–58. http://dx.doi.org/10.1016/j.endm.2015.05.038.

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3

Carocca, Angel, Herbert Lange, and Rubí E. Rodríguez. "Abelian varieties with finite abelian group action." Archiv der Mathematik 112, no. 6 (April 20, 2019): 615–22. http://dx.doi.org/10.1007/s00013-018-1291-9.

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4

K. Thomas, Eldho, Nadya Markin, and Frédérique Oggier. "On Abelian group representability of finite groups." Advances in Mathematics of Communications 8, no. 2 (2014): 139–52. http://dx.doi.org/10.3934/amc.2014.8.139.

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5

A. Zain, Adnan. "On Group Codes Over Elementary Abelian Groups." Sultan Qaboos University Journal for Science [SQUJS] 8, no. 2 (June 1, 2003): 145. http://dx.doi.org/10.24200/squjs.vol8iss2pp145-151.

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For group codes over elementary Abelian groups we present definitions of the generator and the parity check matrices, which are matrices over the ring of endomorphism of the group. We also lift the theorem that relates the parity check and the generator matrices of linear codes over finite fields to group codes over elementary Abelian groups. Some new codes that are MDS, self-dual, and cyclic over the Abelian group with four elements are given.

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6

Carocca, Angel, Herbert Lange, and Rubí E. Rodríguez. "RETRACTED ARTICLE: Abelian varieties with finite abelian group action." Archiv der Mathematik 112, no. 4 (October 8, 2018): 447–48. http://dx.doi.org/10.1007/s00013-018-1244-3.

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7

Li, Xiaolong, Siren Cai, Weiming Zhang, and Bin Yang. "Matrix embedding in finite abelian group." Signal Processing 113 (August 2015): 250–58. http://dx.doi.org/10.1016/j.sigpro.2015.02.007.

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8

Gao, Yubin, and Guoping Tang. "K2 of finite abelian group algebras." Journal of Pure and Applied Algebra 213, no. 7 (July 2009): 1201–7. http://dx.doi.org/10.1016/j.jpaa.2008.09.015.

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9

PILLADO, CRISTINA GARCÍA, SANTOS GONZÁLEZ, CONSUELO MARTÍNEZ, VICTOR MARKOV, and ALEXANDER NECHAEV. "GROUP CODES OVER NON-ABELIAN GROUPS." Journal of Algebra and Its Applications 12, no. 07 (May 16, 2013): 1350037. http://dx.doi.org/10.1142/s0219498813500370.

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Let G be a finite group and F a field. We show that all G-codes over F are abelian if the order of G is less than 24, but for F = ℤ5 and G = S4 there exist non-abelian G-codes over F, answering to an open problem posed in [J. J. Bernal, Á. del Río and J. J. Simón, An intrinsical description of group codes, Des. Codes Cryptogr.51(3) (2009) 289–300]. This problem is related to the decomposability of a group as the product of two abelian subgroups. We consider this problem in the case of p-groups, finding the minimal order for which all p-groups of such order are decomposable. Finally, we study if the fact that all G-codes are abelian remains true when the base field is changed.

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10

Han, Dongchun, Yuan Ren, and Hanbin Zhang. "On ∗-clean group rings over abelian groups." Journal of Algebra and Its Applications 16, no. 08 (August 9, 2016): 1750152. http://dx.doi.org/10.1142/s0219498817501523.

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An associative ring with unity is called clean if each of its elements is the sum of an idempotent and a unit. A clean ring with involution ∗ is called ∗-clean if each of its elements is the sum of a unit and a projection (∗-invariant idempotent). In a recent paper, Huang, Li and Yuan provided a complete characterization that when a group ring [Formula: see text] is ∗-clean, where [Formula: see text] is a finite field and [Formula: see text] is a cyclic group of an odd prime power order [Formula: see text]. They also provided a necessary condition and a few sufficient conditions for [Formula: see text] to be ∗-clean, where [Formula: see text] is a cyclic group of order [Formula: see text]. In this paper, we extend the above result of Huang, Li and Yuan from [Formula: see text] to [Formula: see text] and provide a characterization of ∗-clean group rings [Formula: see text], where [Formula: see text] is a finite abelian group and [Formula: see text] is a field with characteristic not dividing the exponent of [Formula: see text].

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11

Gupta, Shalini. "Finite Metabelian Group Algebras." International Journal of Pure Mathematical Sciences 17 (October 2016): 30–38. http://dx.doi.org/10.18052/www.scipress.com/ijpms.17.30.

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Given a finite metabelian group G, whose central quotient is abelian (not cyclic) group of order p2, p odd prime, the objective of this paper is to obtain a complete algebraic structure of semisimple group algebra Fq[G] in terms of primitive central idempotents, Wedderburn decomposition and the automorphism group.

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12

Rybakov, Sergey. "Finite group subschemes of abelian varieties over finite fields." Finite Fields and Their Applications 29 (September 2014): 132–50. http://dx.doi.org/10.1016/j.ffa.2014.04.001.

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13

Deaconescu, Marian, and Gheorghe Silberberg. "Finite co-Dedekindian groups." Glasgow Mathematical Journal 38, no. 2 (May 1996): 163–69. http://dx.doi.org/10.1017/s0017089500031396.

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A group G is called Dedekindian if every subgroup ofG is normal in G.The structure of the finite Dedekindian groups is well-known [3, Satz 7.12]. They are either abelian or direct products of the form Q × A × B, where Q is the quaternion group of order 8, Ais abelian of odd order and exp(B) ≤ 2.

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14

WEHRFRITZ, B. A. F. "FINITE-FINITARY GROUPS OF AUTOMORPHISMS." Journal of Algebra and Its Applications 01, no. 04 (December 2002): 375–89. http://dx.doi.org/10.1142/s0219498802000318.

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In this paper we attempt to describe the structure of groups G of automorphisms of an abelian group M with the property that M(g - 1) is finite for every element g of G. These groups are closely related to the finitary linear groups over finite fields. The abelian case is critical for our work and the core result of this paper is the following. An abelian group A is isomorphic to a group G as above with M torsion if and only if A is torsion and has a residually-finite subgroup B with A/B a direct sum of cyclic groups.

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15

El Badry, Mohammed, Mostafa Alaoui Abdallaoui, and Abdelfattah Haily. "Infinite groups whose group algebras satisfy the converse of Schur’s lemma." Journal of Algebra and Its Applications 18, no. 10 (August 6, 2019): 1950186. http://dx.doi.org/10.1142/s021949881950186x.

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In this work, we give some necessary and/or sufficient conditions for a group algebra of infinite group to satisfy the converse of Schur’s Lemma. Many classes of groups are investigated such as abelian groups, hypercentral groups, groups having abelian subgroup of finite index and finitely generated soluble groups.

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16

Deaconescu, M., R. R. Khazal, and G. L. Walls. "FORCING A FINITE GROUP TO BE ABELIAN." Mathematical Proceedings of the Royal Irish Academy 111, no. -1 (January 1, 2011): 29–34. http://dx.doi.org/10.3318/pria.2011.111.1.4.

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17

Guo, Shu-Guang. "Restricted sumsets in a finite abelian group." Discrete Mathematics 309, no. 23-24 (December 2009): 6530–34. http://dx.doi.org/10.1016/j.disc.2009.06.028.

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18

Navarro, Gabriel, and Pham Huu Tiep. "Abelian Sylow subgroups in a finite group." Journal of Algebra 398 (January 2014): 519–26. http://dx.doi.org/10.1016/j.jalgebra.2013.04.007.

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19

Evans, Anthony B. "On Elementary Abelian Cartesian Groups." Canadian Mathematical Bulletin 34, no. 1 (March 1, 1991): 58–59. http://dx.doi.org/10.4153/cmb-1991-009-3.

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AbstractJ. Hayden [2] proved that, if a finite abelian group is a Cartesian group satisfying a certain "homogeneity condition", then it must be an elementary abelian group. His proof required the character theory of finite abelian groups. In this note we present a shorter, elementary proof of his result.

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20

Quackenbush, R., and C. S. Szabó. "Nilpotent groups are not dualizable." Journal of the Australian Mathematical Society 72, no. 2 (April 2002): 173–80. http://dx.doi.org/10.1017/s1446788700003827.

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AbstractIt is shown that no finite group containing a non-abelian nilpotent subgroup is dualizable. This is in contrast to the known result that every finite abelian group is dualizable (as part of the Pontryagin duality for all abelian groups) and to the result of the authors in a companion article that every finite group with cyclic Sylow subgroups is dualizable.

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21

Wehrfritz, B. A. F. "ON HYPER AND HYPO ABELIAN-OF-FINITE-RANK GROUPS." Asian-European Journal of Mathematics 01, no. 03 (September 2008): 431–38. http://dx.doi.org/10.1142/s1793557108000369.

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22

Gluck, David. "The largest Irreducible Character Degree of a Finite Group." Canadian Journal of Mathematics 37, no. 3 (June 1, 1985): 442–51. http://dx.doi.org/10.4153/cjm-1985-026-8.

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Much information about a finite group is encoded in its character table. Indeed even a small portion of the character table may reveal significant information about the group. By a famous theorem of Jordan, knowing the degree of one faithful irreducible character of a finite group gives an upper bound for the index of its largest normal abelian subgroup.Here we consider b(G), the largest irreducible character degree of the group G. A simple application of Frobenius reciprocity shows that b(G) ≧ |G:A| for any abelian subgroup A of G. In light of this fact and Jordan's theorem, one might seek to bound the index of the largest abelian subgroup of G from above by a function of b(G). If is G is nilpotent, a result of Isaacs and Passman (see [7, Theorem 12.26]) shows that G has an abelian subgroup of index at most b(G)4.

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23

Tushev, A. V. "On certain methods of studying ideals in group rings of abelian groups of finite rank." Asian-European Journal of Mathematics 07, no. 04 (December 2014): 1450065. http://dx.doi.org/10.1142/s179355711450065x.

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Over the last ten years a deep theory of structure of group rings of abelian groups of finite ring was developed mostly due to techniques based on valuation theory of fields. However, as it becomes clear in the beginning of the century, there are strong restrictions for applying such techniques. We considered certain new methods available for studying group rings of abelian groups of finite rank.

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24

Hertweck, Martin. "Unit Groups of Integral Finite Group Rings with No Noncyclic Abelian Finitep-Subgroups." Communications in Algebra 36, no. 9 (September 17, 2008): 3224–29. http://dx.doi.org/10.1080/00927870802103669.

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25

BOYLE, MIKE, and SCOTT SCHMIEDING. "Finite group extensions of shifts of finite type: -theory, Parry and Livšic." Ergodic Theory and Dynamical Systems 37, no. 4 (February 11, 2016): 1026–59. http://dx.doi.org/10.1017/etds.2015.87.

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This paper extends and applies algebraic invariants and constructions for mixing finite group extensions of shifts of finite type. For a finite abelian group$G$, Parry showed how to define a$G$-extension$S_{A}$from a square matrix over$\mathbb{Z}_{+}G$, and classified the extensions up to topological conjugacy by the strong shift equivalence class of$A$over$\mathbb{Z}_{+}G$. Parry asked, in this case, if the dynamical zeta function$\det (I-tA)^{-1}$(which captures the ‘periodic data’ of the extension) would classify the extensions by$G$of a fixed mixing shift of finite type up to a finite number of topological conjugacy classes. When the algebraic$\text{K}$-theory group$\text{NK}_{1}(\mathbb{Z}G)$is non-trivial (e.g. for$G=\mathbb{Z}/n$with$n$not square-free) and the mixing shift of finite type is not just a fixed point, we show that the dynamical zeta function for any such extension is consistent with an infinite number of topological conjugacy classes. Independent of$\text{NK}_{1}(\mathbb{Z}G)$, for every non-trivial abelian$G$we show that there exists a shift of finite type with an infinite family of mixing non-conjugate$G$extensions with the same dynamical zeta function. We define computable complete invariants for the periodic data of the extension for$G$(not necessarily abelian), and extend all the above results to the non-abelian case. There is other work on basic invariants. The constructions require the ‘positive$K$-theory’ setting for positive equivalence of matrices over$\mathbb{Z}G[t]$.

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26

Außenhofer, L., and S. S. Gabriyelyan. "On reflexive group topologies on abelian groups of finite exponent." Archiv der Mathematik 99, no. 6 (November 22, 2012): 583–88. http://dx.doi.org/10.1007/s00013-012-0455-2.

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27

Bettio, Egle. "Groups of finite exponent acting regulary on an abelian group." Archiv der Mathematik 106, no. 3 (February 6, 2016): 219–23. http://dx.doi.org/10.1007/s00013-016-0870-x.

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28

Nesin, Ali. "Groups of finite Morley rank with transitive group automorphisms." Journal of Symbolic Logic 54, no. 3 (September 1989): 1080–82. http://dx.doi.org/10.2307/2274766.

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The aim of this short note is to prove the following result:Theorem. Let G be a group of finite Morley rank with Aut G acting transitively on G/{1}. Then G is either abelian or a bad group.Bad groups were first defined by Cherlin [Ch]: these are groups of finite Morley rank without solvable and nonnilpotent connected subgroups. They have been investigated by the author [Ne 1], Borovik [Bo], Corredor [Co], and Poizat and Borovik [Bo-Po]. They are not supposed to exist, but we are far from proving their nonexistence. This is one of the major obstacles to proving Cherlin's conjecture: infinite simple groups of finite Morley rank are algebraic groups.If the group G of the theorem is finite, then it is well known that G ≈ ⊕Zp for some prime p: clearly all elements of G have the same order, say p, a prime. Thus G is a finite p-group, so has a nontrivial center. But Aut G acts transitively; thus G is abelian. Since it has exponent p, G ≈ ⊕Zp.The same proof for infinite G does not work even if it has finite Morley rank, for the following reasons:1) G may not contain an element of finite order.2) Even if G does contain an element of finite order, i.e. if G has exponent p, we do not know if G must have a nontrivial center.

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29

Han, Lingling, and Xiuyun Guo. "The Number of Subgroup Chains of Finite Nilpotent Groups." Symmetry 12, no. 9 (September 17, 2020): 1537. http://dx.doi.org/10.3390/sym12091537.

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In this paper, we mainly count the number of subgroup chains of a finite nilpotent group. We derive a recursive formula that reduces the counting problem to that of finite p-groups. As applications of our main result, the classification problem of distinct fuzzy subgroups of finite abelian groups is reduced to that of finite abelian p-groups. In particular, an explicit recursive formula for the number of distinct fuzzy subgroups of a finite abelian group whose Sylow subgroups are cyclic groups or elementary abelian groups is given.

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30

Chiang, Li, and Shi-Shyr Roan. "Orbifolds and finite group representations." International Journal of Mathematics and Mathematical Sciences 26, no. 11 (2001): 649–69. http://dx.doi.org/10.1155/s0161171201020154.

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We present our recent understanding on resolutions of Gorenstein orbifolds, which involves the finite group representation theory. We concern only the quotient singularity of hypersurface type. The abelian groupAr(n)forA-type hypersurface quotient singularity of dimensionnis introduced. Forn=4, the structure of Hilbert scheme of group orbits and crepant resolutions ofAr(4)-singularity are obtained. The flop procedure of4-folds is explicitly constructed through the process.

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31

Morris, A. O., M. Saeed-Ul-Islam, and E. Thomas. "Some projective representations of finite abelian groups." Glasgow Mathematical Journal 29, no. 2 (July 1987): 197–203. http://dx.doi.org/10.1017/s0017089500006844.

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In this paper, we continue the work initiated by Morris [5] and Saeed-ul-Islam [6,7] and determine complete sets of inequivalent irreducible projective representations (which we shall write as i.p.r.) of finite Abelian groups with respect to some additional factor sets.We consider an Abelian groupwhich will be referred to as an Abelian group of type (a1, …, am).

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32

Steinberg, Benjamin. "Monoid kernels and profinite topologies on the free Abelian group." Bulletin of the Australian Mathematical Society 60, no. 3 (December 1999): 391–402. http://dx.doi.org/10.1017/s0004972700036571.

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To each pseudovariety of Abelian groups residually containing the integers, there is naturally associated a profinite topology on any finite rank free Abelian group. We show in this paper that if the pseudovariety in question has a decidable membership problem, then one can effectively compute membership in the closure of a subgroup and, more generally, in the closure of a rational subset of such a free Abelian group. Several applications to monoid kernels and finite monoid theory are discussed.

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33

Dokuchaev, Michael A., and Stanley O. Juriaans. "Finite Subgroups in Integral Group Rings." Canadian Journal of Mathematics 48, no. 6 (December 1, 1996): 1170–79. http://dx.doi.org/10.4153/cjm-1996-061-7.

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AbstractA p-subgroup version of the conjecture of Zassenhaus is proved for some finite solvable groups including solvable groups in which any Sylow p-subgroup is either abelian or generalized quaternion, solvable Frobenius groups, nilpotent-by-nilpotent groups and solvable groups whose orders are not divisible by the fourth power of any prime.

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34

Angiono, Iván, and César Galindo. "Pointed finite tensor categories over abelian groups." International Journal of Mathematics 28, no. 11 (October 2017): 1750087. http://dx.doi.org/10.1142/s0129167x17500872.

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We give a characterization of finite pointed tensor categories obtained as de-equivariantizations of the category of corepresentations of finite-dimensional pointed Hopf algebras with abelian group of group-like elements only in terms of the (cohomology class of the) associator of the pointed part. As an application we prove that every coradically graded pointed finite braided tensor category is a de-equivariantization of the category of corepresentations of a finite-dimensional pointed Hopf algebras with abelian group of group-like elements.

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35

Arai, Kenichi, Hiroyuki Okazaki, and Yasunari Shidama. "Isomorphisms of Direct Products of Finite Cyclic Groups." Formalized Mathematics 20, no. 4 (December 1, 2012): 343–47. http://dx.doi.org/10.2478/v10037-012-0038-5.

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Summary In this article, we formalize that every finite cyclic group is isomorphic to a direct product of finite cyclic groups which orders are relative prime. This theorem is closely related to the Chinese Remainder theorem ([18]) and is a useful lemma to prove the basis theorem for finite abelian groups and the fundamental theorem of finite abelian groups. Moreover, we formalize some facts about the product of a finite sequence of abelian groups.

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36

Broughton, Allen, and Aaron Wootton. "Finite abelian subgroups of the mapping class group." Algebraic & Geometric Topology 7, no. 4 (December 17, 2007): 1651–97. http://dx.doi.org/10.2140/agt.2007.7.1651.

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37

Cherly, Jorgen. "An Addition Theorem in a Finite Abelian Group." MATHEMATICA SCANDINAVICA 74 (June 1, 1994): 5. http://dx.doi.org/10.7146/math.scand.a-12474.

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38

Szőnyi, Tamás, and Ferenc Wettl. "On complexes in a finite abelian group, I." Proceedings of the Japan Academy, Series A, Mathematical Sciences 64, no. 7 (1988): 245–48. http://dx.doi.org/10.3792/pjaa.64.245.

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39

Szőnyi, Tamás, and Ferenc Wettl. "On complexes in a finite abelian group, II." Proceedings of the Japan Academy, Series A, Mathematical Sciences 64, no. 8 (1988): 286–87. http://dx.doi.org/10.3792/pjaa.64.286.

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40

Buchmann, Johannes, and Arthur Schmidt. "Computing the structure of a finite abelian group." Mathematics of Computation 74, no. 252 (March 8, 2005): 2017–27. http://dx.doi.org/10.1090/s0025-5718-05-01740-0.

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41

Benwen, Huang. "Finite Abelian group of |A(G)|=27 pq." Wuhan University Journal of Natural Sciences 1, no. 1 (March 1996): 25–30. http://dx.doi.org/10.1007/bf02827573.

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42

Gao, W., Y. O. Hamidoune, A. Lladó*, and O. Serra†. "Covering a Finite Abelian Group by Subset Sums." COMBINATORICA 23, no. 4 (December 2003): 599–611. http://dx.doi.org/10.1007/s00493-003-0036-x.

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43

Navarro, Gabriel, Ronald Solomon, and Pham Huu Tiep. "Abelian Sylow subgroups in a finite group, II." Journal of Algebra 421 (January 2015): 3–11. http://dx.doi.org/10.1016/j.jalgebra.2014.08.012.

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44

Kurdachenko, L. A., and N. N. Semko. "On the structure of some groups having finite contranormal subgroups." Algebra and Discrete Mathematics 31, no. 1 (2021): 109–19. http://dx.doi.org/10.12958/adm1724.

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Following J.S. Rose, a subgroup H of the group G is said to be contranormal in G, if G=HG. In a certain sense, contranormal subgroups are antipodes to subnormal subgroups. We study the structure of Abelian-by-nilpotent groups having a finite proper contranormal p-subgroup.

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45

THOMAS, VIJI Z. "THE NON-ABELIAN TENSOR PRODUCT OF FINITE GROUPS IS FINITE: A HOMOLOGY-FREE PROOF." Glasgow Mathematical Journal 52, no. 3 (August 25, 2010): 473–77. http://dx.doi.org/10.1017/s0017089510000352.

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AbstractIn this note, we give a homology-free proof that the non-abelian tensor product of two finite groups is finite. In addition, we provide an explicit proof that the non-abelian tensor product of two finite p-groups is a finite p-group.

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46

Bertram, Edward A., and Marcel Herzog. "Finite groups with large centralizers." Bulletin of the Australian Mathematical Society 32, no. 3 (December 1985): 399–414. http://dx.doi.org/10.1017/s0004972700002513.

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It is known that a finite non-abelian group G has a proper centralizer of order if, for example, |G| is even and |Z(G)| is odd, or whenever G is solvable. Often the exponent can be improved to , for example when G is supersolvable, or metabelian, or |G = pαqβ. Here we show more generally that this improvement is possible in many situations where G is factorizable into the product of two subgroups. In particular, much more evidence is presented to support the conjecture that some proper centralizer has order whenever G is a finite non-abelian solvable group.

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47

Banakh, T. O., and V. M. Gavrylkiv. "Bases in finite groups of small order." Carpathian Mathematical Publications 13, no. 1 (June 20, 2021): 149–59. http://dx.doi.org/10.15330/cmp.13.1.149-159.

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A subset $B$ of a group $G$ is called a basis of $G$ if each element $g\in G$ can be written as $g=ab$ for some elements $a,b\in B$. The smallest cardinality $|B|$ of a basis $B\subseteq G$ is called the basis size of $G$ and is denoted by $r[G]$. We prove that each finite group $G$ has $r[G]>\sqrt{|G|}$. If $G$ is Abelian, then $r[G]\ge \sqrt{2|G|-|G|/|G_2|}$, where $G_2=\{g\in G:g^{-1} = g\}$. Also we calculate the basis sizes of all Abelian groups of order $\le 60$ and all non-Abelian groups of order $\le 40$.

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48

Kurdachenko, L. A., A. A. Pypka, and I. Ya Subbotin. "On analogs of some classical group-theoretic results in Poisson algebras." Reports of the National Academy of Sciences of Ukraine, no. 3 (July 6, 2021): 11–16. http://dx.doi.org/10.15407/dopovidi2021.03.011.

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We investigate the Poisson algebras, in which the n-th hypercenter (center) has a finite codimension. It was established that, in this case, the Poisson algebra P includes a finite-dimensional ideal K such that P/K is nilpotent (Abelian). Moreover, if the n-th hypercenter of a Poisson algebra P over some field has a finite codimension, and if P does not contain zero divisors, then P is Abelian.

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Erfanian, A., and M. Farrokhi D.G. "Finite Groups with Four Relative Commutativity Degrees." Algebra Colloquium 22, no. 03 (July 14, 2015): 449–58. http://dx.doi.org/10.1142/s1005386715000401.

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Abstract:

It is shown that a finite group G has four relative commutativity degrees if and only if G/Z(G) is a p-group of order p3 and G has no abelian maximal subgroups, or G/Z(G) is a Frobenius group with Frobenius kernel and complement isomorphic to ℤp × ℤp and ℤq, respectively, and the Sylow p-subgroup of G is abelian, where p and q are distinct primes.

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HADAD, UZY. "KAZHDAN CONSTANTS OF GROUP EXTENSIONS." International Journal of Algebra and Computation 20, no. 05 (August 2010): 671–88. http://dx.doi.org/10.1142/s0218196710005832.

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Abstract:

We give bounds on Kazhdan constants of abelian extensions of (finite) groups. As a corollary, we improved known results of Kazhdan constants for some meta-abelian groups and for the relatively free group in the variety of p-groups of lower p-series of class 2. Furthermore, we calculate Kazhdan constants of the tame automorphism groups of the free nilpotent groups.

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