Hypercube problems Lecture 3 October 24, 2012
Lecturer: Petr Gregor Scribe by: Otakar Trunda Updated: November 8, 2012
1 Structure of the automorphism group
In this talk we take a closer look at the structure of Γ = Aut(Qn). We already know that
n ∀g ∈ Γ ∃!π ∈ Sn ∃!a ∈ Z2 such that g = rπta where rπ ∈ Rn, ta ∈ Tn.
A subgroup N of a group G is normal (denoted by N C G) if it is invariant under −1 conjugation; that is, gNg = N for every g ∈ G. It is easy to see that Tn C Γ for every n ≥ 1 and Rn 6 Γ for every n ≥ 2. Definition 1 Let G, H be groups. The direct product of G and H (denoted by G × H) is the group on {(g, h) | g ∈ G, h ∈ H} with the operation defined by (g1, h1)(g2, h2) = (g1g2, h1h2).
Note that Γ cannot be expressed as Rn ×Tn since (rπ, ta)(rρ, tb) 6= (rπrρ, tatb). However, we can use a conjugation to express the composition since
(rπ, ta)(rρ, tb) = (rπ(rρrρ−1 ), ta)(rρ, tb) = (rπrρ, (rρ−1 tarρ)tb).
Definition 2 Let H, N be groups and let ϕ : H → Aut(N) be a homomorphism. The semidirect product (external) of H and N with respect to ϕ (we write H nϕ N) is the group on −1 {(h, n), h ∈ H, n ∈ N} with the operation defined by (h1, n1)(h2, n2) = (h1h2, ϕ(h2 )(n1)n2).
If H, N are subgroups of same group and N is normal then we can use the homomorphism −1 ϕ defined by ϕ(h): n 7→ hnh (a conjugation by h). Then we write simply H n N.
n Fact 3 Aut(Qn) = Rn n Tn ' Sn n Z2 for every n ≥ 1.
To verify that Aut(Qn) can be expressed in this way, let us see the composition
−1 −1 (rπ, ta)(rρ, tb) = (rπrρ, ϕ(rρ )(ta)tb) = (rπrρ, rρ tarρtb) = (rπrρ, taρ tb) = (rπρ, taρ⊕b)
−1 since rρ tarρ : u 7→ (uρ−1 ⊕ a)ρ = u ⊕ aρ. The neutral element is (rid, t0) and the inverse is
−1 −1 −1 −1 −1 −1 −1 −1 −1 (r , t ) = (r , [ϕ(r )(t )] ) = (r , [r t r ] ) = (r , r t r ) = (r −1 , t ). π a π π a π π a π π π a π π aπ−1
Aut(Qn) is called the hyperoctahedral group and it is also the group of symmetries of a cross-polytope (a dual polytope to the hypercube). Alternatively, Aut(Qn) can be written as the wreath product Aut(Qn) ' S2 o Sn, see the following definition.
3-1 Definition 4 Let A, H be groups, H acting on V . Let K be the direct product of copies Q of A indexed by elements of V (i.e. K = x∈V Ax, K is called a base.) The (unrestricted) wreath product of A and H (denoted by A oV H) is A oV H = H n K.
If H = Sn we take V = [n] with the natural action of H on V and write simply A o H.
In the first lecture we defined the folded cube FQn and the augmented cube AQn. It is n n+3 known that Aut(FQn) ' Sn+1 n Z2 [8] and |Aut(AQn)| = 2 [5].
Problem 1 What is the structure of Aut(AQn)?
2 Distance transitivity
In this section we inspect how Aut(Qn) acts on ordered pairs of vertices. Definition 5 Let Γ be a group acting on V and let x, y ∈ V . The orbital of (x, y) is a set
Γ(x, y) = {(g(x), g(y)) | g ∈ Γ}.
In fact, it is an orbit in Γ with the action on V × V induced by the action on V .
Let n ≥ 1 be fixed, Γ = Aut(Qn) with the action on V = V (Qn). For 0 ≤ d ≤ n let
Dd = {(u, v) ∈ V × V | dH (u, v) = d}.