Bardakov, Valeriy G.; Bryukhanov, Oleg V.; Gongopadhyay, Krishnendu Palindromic widths of nilpotent and wreath products. (English) Zbl 1365.20024 Proc. Indian Acad. Sci., Math. Sci. 127, No. 1, 99-108 (2017). Summary: We prove that the nilpotent product of a set of groups \(A_{1},\dotsc,A_{s}\) has finite palindromic width if and only if the palindromic widths of \(A_{i},\,i=1,\dotsc,s\), are finite. We give a new proof that the commutator width of \(F_{n}\wr K\) is infinite, where \(F_{n}\) is a free group of rank \(n\geq 2\) and \(K\) is a finite group. This result, combining with a result of E. Fink [J. Algebra 471, 1–12 (2017; Zbl 1371.20030)] gives examples of groups with infinite commutator width but finite palindromic width with respect to some generating set. Cited in 2 Documents MSC: 20E22 Extensions, wreath products, and other compositions of groups 20F16 Solvable groups, supersolvable groups 20F05 Generators, relations, and presentations of groups 20F19 Generalizations of solvable and nilpotent groups Keywords:palindromic width; commutator width; wreath products; nilpotent product Citations:Zbl 1371.20030 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Akhavan-Malayeri M, On commutator length and square length of the wreath product of a group by a finitely generated abelian group, Algebra Colloquium17(Spec 1) (2010) 799-802 · Zbl 1203.20034 · doi:10.1142/S100538671000074X [2] Bardakov V, On the theory of braid groups, Russian Acad. Sci., Sb., Math.76(1) (1993) 123-153, translation from Mat. Sb.183(6) (1992) 3-42 · Zbl 0798.20029 [3] Bardakov V, Computation of commutator length in free groups, (Russian) Algebra i Logika39(4) (2000) 395-440, translation in Algebra and Logic39(4) (2000) 224-251 · Zbl 0960.20019 [4] Bardakov V G and Gongopadhyay K, Palindromic width of free nilpotent groups, J. Algebra402 (2014) 379-391 · Zbl 1308.20034 · doi:10.1016/j.jalgebra.2013.12.002 [5] Bardakov V G and Gongopadhyay K, On palindromic width of certain extensions and quotients of free nilpotent groups, Int. J. Algebra Comput.24(5) (2014) 553-567 · Zbl 1354.20021 · doi:10.1142/S0218196714500246 [6] Bardakov V G and Gongopadhyay K, Palindromic width of finitely generated solvable groups, Comm. Algebra43(11) (2015) 4809-4824 · Zbl 1344.20044 · doi:10.1080/00927872.2014.952738 [7] Bardakov V, Shpilrain V and Tolstykh V, On the palindromic and primitive widths of a free group, J. Algebra285 (2005) 574-585 · Zbl 1085.20011 · doi:10.1016/j.jalgebra.2004.11.003 [8] Bardakov V and Tolstykh V, The palindromic width of a free product of groups, J. Aust. Math. Soc.81(2) (2006) 199-208 · Zbl 1114.20012 · doi:10.1017/S1446788700015822 [9] Fink E, Palindromic width of wreath products, arXiv 1402.4345 · Zbl 1371.20030 [10] Fink E and Thom A, Palindromic words in simple groups, Internat. J. Algebra Comput.25(3) (2015) 439-444 · Zbl 1321.20028 · doi:10.1142/S0218196715500046 [11] Fink E, Conjugacy growth and conjugacy width of certain branch groups, Int. J. Algebra Comput.24(8) (2014) 1213-1232 · Zbl 1318.20028 · doi:10.1142/S0218196714500544 [12] Golovin O N, Nilpotentnii Proiszvedeniya Grup (Russian), Mat. Sbornik27(3) (1950) 427-454; Nilpotent products of groups, Amer. Math. Soc. Transl.2(2) (1956) 89-115 · Zbl 0070.25401 [13] Golovin O N, The metabelian products of groups, (Russian) Mat. Sb., N. Ser.28(70) (1951) 431-444 · Zbl 0042.01803 [14] Golovin O N, On the problem of isomorphisms of nilpotent decompositions of a group, (English) Am. Math. Soc., Transl., II. Ser.2 (1956) 133-145 · Zbl 0070.25403 · doi:10.1090/trans2/002/05 [15] Moran S, Associative operations on groups, I, Proc. London Math. Soc.6(3) (1956) 581-596 · Zbl 0073.01202 · doi:10.1112/plms/s3-6.4.581 [16] Moran S, Associative operations on groups, II, Proc. London Math. Soc.8(3) (1958) 548-568 · Zbl 0097.25303 · doi:10.1112/plms/s3-8.4.548 [17] Moran S, Associative operations on groups, III, Proc. London Math. Soc.9(3) (1959) 287-317 · doi:10.1112/plms/s3-9.2.287 [18] Nikolov N, On the commutator width of perfect groups, Bull. London Math. Soc.36(1) (2004) 30-36 · Zbl 1048.20013 · doi:10.1112/S0024609303002601 [19] Piggott A, Palindromic primitives and palindromic bases in the free group of rank two, J. Algebra304 (2006) 359-366 · Zbl 1111.20028 · doi:10.1016/j.jalgebra.2005.12.005 [20] Rhemtulla A H, A problem of bounded expressibility in free groups, Math. Proc. Cambridge Philos. Soc.64 (1969) 573-584 · Zbl 0159.03001 · doi:10.1017/S0305004100043231 [21] Riley T R and Sale A W, Palindromic width of wreath products, metabelian groups and max-n solvable groups, Groups Complexity Cryptology6(2) (2014) 121-132 · Zbl 1312.20034 · doi:10.1515/gcc-2014-0009 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.