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Keywords:
loop; elementary abelian group; inner mapping group
Summary:
We show that finite commutative inverse property loops with elementary abelian inner mapping groups of order $p^5$ are centrally nilpotent of class at most two.
References:
[1] Bruck R. H.: Contributions to the theory of loops. Trans. Amer. Math. Soc. 60 (1946), 245–354. DOI 10.1090/S0002-9947-1946-0017288-3 | MR 0017288 | Zbl 0061.02201
[2] Csörgö P.: On connected transversals to abelian subgroups and loop theoretical consequences. Arch. Math. (Basel) 86 (2006), no. 6, 499–516. DOI 10.1007/s00013-006-1379-5 | MR 2241599
[3] Csörgö P.: Abelian inner mappings and nilpotency class greater than two. European J. Combin. 28 (2007), no. 3, 858–867. DOI 10.1016/j.ejc.2005.12.002 | MR 2300766 | Zbl 1149.20053
[4] Drápal A., Vojtěchovský P.: Explicit constructions of loops with commuting inner mappings. European J. Combin. 29 (2008), no. 7, 1662–1681. DOI 10.1016/j.ejc.2007.10.001 | MR 2431758
[5] Leppälä E., Niemenmaa M.: On finite commutative loops which are centrally nilpotent. Comment. Math. Univ. Carolin. 56 (2015), no. 2, 139–143. MR 3338728 | Zbl 1339.20064
[6] Niemenmaa M.: Finite loops with nilpotent inner mapping groups are centrally nilpotent. Bull. Aust. Math. Soc. 79 (2009), no. 1, 109–114. DOI 10.1017/S0004972708001093 | MR 2486887 | Zbl 1167.20039
[7] Niemenmaa M.: On finite commutative IP-loops with elementary abelian inner mapping groups of order $p^4$. Comment. Math. Univ. Carolin. 51 (2010), no. 4, 559–563. MR 2858260
[8] Niemenmaa M.: On dihedral $2$-groups as inner mapping groups of finite commutative inverse property loops. Comment. Math. Univ. Carolin. 59 (2018), no. 2, 189–193. MR 3815684
[9] Niemenmaa M., Kepka T.: On multiplication groups of loops. J. Algebra 135 (1990), no. 1, 112–122. DOI 10.1016/0021-8693(90)90152-E | MR 1076080 | Zbl 0706.20046
[10] Niemenmaa M., Kepka T.: On connected transversals to abelian subgroups. Bull. Austral. Math. Soc. 49 (1994), no. 1, 121–128. DOI 10.1017/S0004972700016166 | MR 1262682 | Zbl 0799.20020
[11] Niemenmaa M., Rytty M.: Connected transversals and multiplication groups of loops. Quasigroups Related Systems 15 (2007), no. 1, 95–107. MR 2379127 | Zbl 1133.20009
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