### On the structure of finite loop capable nilpotent groups

Miikka Rytty (2010)

Commentationes Mathematicae Universitatis Carolinae

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In this paper we consider finite loops and discuss the problem which nilpotent groups are isomorphic to the inner mapping group of a loop. We recall some earlier results and by using connected transversals we transform the problem into a group theoretical one. We will get some new answers as we show that a nilpotent group having either ${C}_{{p}^{k}}\times {C}_{{p}^{l}}$, $k>l\ge 0$ as the Sylow $p$-subgroup for some odd prime $p$ or the group of quaternions as the Sylow $2$-subgroup may not be loop capable.