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Keywords:
group; subgroup; connected transversals; core
group; subgroup; connected transversals; core
Summary:
In this short note, it is shown that if $A,B$ are $H$-connected transversals for a finite subgroup $H$ of an infinite group $G$ such that the index of $H$ in $G$ is at least 3 and $H\cap H^u\cap H^v=1$ whenever $u,v\in G\setminus H$ and $uv^{-1}\in G\setminus H$ then $A=B$ is a normal abelian subgroup of $G$.
References:
[1] Drápal A.: Multiplication groups of free loops I. Czech. Math. J. 46 (121) (1996), 121-131. MR 1371694
[2] Drápal A.: Multiplication groups of free loops II. Czech. Math. J. 46 (121) (1996), 201-220. MR 1388610
[3] Drápal A.: Multiplication groups of finite loops that fix at most two points. submitted.
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