Let K be a finite, connected, abstract simplicial complex. The Morse complex of K, first introduced by Chari and Joswig, is the constructed from all gradient vector fields on K. We show that if neither boundary n-simplex nor cycle, then Aut(M(K))≅Aut(K). In case where K=Cn, cycle length n, we Aut(M(Cn))≅Aut(C2n). When K=∂Δn, prove Aut(M(∂Δn))≅Aut(∂Δn)×Z2. These results are based recent work Cap...

If z is an automorphism (or an endomorphism) of H, then it is also an automorphism (or an endomorphism) of %H and of %H. Hence the automorphism group Aut H is a subgroup of Aut 9?H and of Aut %H. Is there any other relation between the groups Aut H, Aut CBH and Aut %H? More in detail, which spans of groups, see Fig. 1, where m,, m2 are monomorphisms, can be realized by directed graphs H (in the...

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We use extensions to study the semi-simple quotient of the group ring FpAut(P ) of a finite p-group P . For an extension E : N → P → Q, our results involve relations between Aut(N), Aut(P ), Aut(Q) and the extension class [E] ∈ H(Q,ZN). One novel feature is the use of the intersection orbit group Ω([E]), defined as the intersection of the orbits Aut(N) · [E] and Aut(Q) · [E] in H(Q,ZN). This gr...

Let n ≥ 2 and Fn be the free group of rank n. Its automorphism group Aut(Fn) has a well-known surjective linear representation ρ : Aut(Fn) −→ Aut(Fn/F ′ n) = GLn(Z) where F ′ n denotes the commutator subgroup of Fn. By Aut (Fn) := ρ(SLn(Z)) we denote the special automorphism group of Fn. For an epimorphism π : Fn → G of Fn onto a finite group G we call Γ(G, π) := {φ ∈ Aut(Fn) | πφ = π} the stan...

The present study characterizes distinctive and specific features of social behavior impairment, termed social behavior profile (SBP), in young males with fragile X syndrome (FraX). Fourteen males with FraX and autism (FraX+Aut), ages 3-8 years, were compared with either 41 FraX boys without autism (Aut), 7 age-matched males with developmental language delay and autism (DLD+Aut), or with 11 boy...

The cross coproduct braided group Aut(C)◮⊳Aut(C) and higher braided group ‘spin chains’. The example of the braided...