2021/11/23 by Stavroula Makri, Makri, Stavroula
Mathematics · #20F28 #20F36 #Advanced Algebra and Geometry #FOS: Mathematics #Finite Group Theory Research #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2111.12134
openalex publication_date 2021/11/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let UVBn and UVPn be the unrestricted virtual braid group and the unrestricted virtual pure braid group on n strands respectively. We study the groups UVBn and UVPn, and our main results are as follows: for n≥ 5, we give a complete description, up to conjugation, to all possible homomorphisms from UVBn to the symmetric group Sn. For n≥ 3, we characterise all possible images of UVBn, under a group homomorphism, to any finite group G. For n≥ 5, we prove that UVPn is a characteristic subgroup of UVBn. In addition, we determine the automorphism group of UVPn and we prove that ℤ2×ℤ2 is a subgroup of the outer automorphism group of UVBn. Lastly, we show that UVBn and UVPn are residually finite and Hopfian but not co-Hopfian. We also remark that some of these results hold accordingly for the welded braid group WBn and we discuss about its automorphism group.