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11.54 (1990)

Solved

Is it true that in the group of coloured braids only the identity braid is a conjugate to its inverse? (For definition see 10.24.)

Contributor: G. S. Makanin

Progress

The answers to both 11.54 and 11.55 are affirmative, since the group $K_n$ of coloured braids is embeddable into the group of those automorphisms of the free group $F_n$ that act trivially modulo the derived subgroup of $F_n$, and this group is residually in the class of torsion-free nilpotent groups, for which the corresponding assertions are true (V. A. Roman’kov, Letter of October, 3, 1990). See also (V. G. Bardakov, Russ. Acad. Sci. Sbornik Math., 76, no. 1 (1993), 123–153).

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