# zbMATH — the first resource for mathematics

On connected transversals to abelian subgroups. (English) Zbl 0799.20020
If $$G$$ is a group, $$H\geq G$$, $$A$$ and $$B$$ are two left transversals to $$H$$ in $$G$$, then we say that $$A$$ and $$B$$ are $$H$$-connected if $$a^{-1} b^{-1}$$ $$ab\in H$$ for every $$a\in A$$ and $$b\in B$$. This concept was introduced by the authors. They prove that if $$H$$ is a finite abelian subgroup in $$G$$ with two $$H$$-connected transversals $$A$$ and $$B$$, then $$G$$ is solvable. If, in addition $$G =\langle A,B\rangle$$ and the core of $$H$$ in $$G$$ is trivial, then $$Z(G)\neq 1$$. They also investigate some special cases where $$H$$ is very close to a cyclic group.
Finally the results are applied to loop theory and it is shown that if the inner mapping group of a finite loop is abelian, then the loop is centrally nilpotent.

##### MSC:
 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks 20N05 Loops, quasigroups 20E07 Subgroup theorems; subgroup growth 20D25 Special subgroups (Frattini, Fitting, etc.)
Full Text:
##### References:
 [1] DOI: 10.1016/0021-8693(68)90050-1 · Zbl 0155.03901 [2] DOI: 10.1016/0021-8693(64)90017-1 · Zbl 0123.01502 [3] DOI: 10.1007/BF01388521 · Zbl 0564.20010 [4] DOI: 10.2307/1990147 · Zbl 0061.02201 [5] DOI: 10.1112/blms/24.4.343 · Zbl 0793.20064 [6] DOI: 10.1016/0021-8693(90)90176-O · Zbl 0718.17028 [7] DOI: 10.1016/0021-8693(90)90152-E · Zbl 0706.20046 [8] DOI: 10.1017/S0305004100067025 · Zbl 0622.20061 [9] Kepka, Bull. Austral. Math. Soc. 38 pp 171– (1988) [10] Huppert, Endliche Gruppen I (1967) · Zbl 0217.07201 [11] Niemenmaa, Groups St.Andrews pp 396– (1991)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.