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 $G$.
@article{119165,
author = {Tom\'a\v s Kepka and Petr N\v emec},
title = {Connected transversals -- the Zassenhaus case},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {41},
year = {2000},
pages = {299-300},
zbl = {1038.20022},
mrnumber = {1780873},
language = {en},
url = {http://dml.mathdoc.fr/item/119165}
}
Kepka, Tomáš; Němec, Petr. Connected transversals -- the Zassenhaus case. Commentationes Mathematicae Universitatis Carolinae, Tome 41 (2000) pp. 299-300. http://gdmltest.u-ga.fr/item/119165/
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