Mahender Singh

Palindromic automorphisms of free groups

Abstract Let F n be the free group of rank n with free basis X = { x 1 , … , x n } . A palindrome is a word in X ± 1 that reads the same backwards as forwards. The palindromic automorphism group Π A n of F n consists of those automorphisms that map each x i to a palindrome. In this paper, we investigate linear representations of Π A n , and prove that Π A 2 is linear. We obtain conjugacy classes of involutions in Π A 2 , and investigate residual nilpotency of Π A n and some of its subgroups. Let IA n be the group of those automorphisms of F n that act trivially on the abelianisation, PI n be the palindromic Torelli group of F n , and let E Π A n be the elementary palindromic automorphism group of F n . We prove that PI n = IA n ∩ E Π A n ′ . This result strengthens a recent result of Fullarton [2] .

Automorphism groups of quandles arising from groups

Let G be a group and

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\varphi \in {\text {Aut}}(G)

General Bourgin–Yang theorems

φ∈Aut(G). Then the set G equipped with the binary operation

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a*b=\varphi (ab^{-1})b

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a∗b=φ(ab-1)b gives a quandle structure on G, denoted by