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Let G be a group with a dihedral subgroup H of order 2pn, where p is an odd prime. We show that if there exist H-connected transversals in G, then G is a solvable group. We apply this result to the loop theory and show that if the inner mapping group of a finite loop Q is dihedral of order 2pn, then Q is a solvable loop.
Let G be a group and t an unknown. In this paper we prove that the equation atbtct−1dt−1 = 1 (a,b,c,d ɛ G, a2 ≠ 1, c2 ≠ 1, bd ≠ 1) has a solution over G. This forms part of a program to investigate precisely when an equation, whose associated star graph contains no admissible paths of length less than 3, fails to have a solution over G.
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