Wreath Product - Examples

Examples

• The Lamplighter group is the restricted wreath product ℤ₂≀ℤ.

• ℤ_m≀S_n (Generalized symmetric group).

The base of this wreath product is the n-fold direct product

ℤ_mn = ℤ_m × ... × ℤ_m

of copies of ℤ_m where the action φ : S_n → Aut(ℤ_mn) of the symmetric group S_n of degree n is given by

φ(σ)(α₁,..., α_n) := (α_σ(1),..., α_σ(n)).

• S₂≀S_n (Hyperoctahedral group).

The action of S_n on {1,...,n} is as above. Since the symmetric group S₂ of degree 2 is isomorphic to ℤ₂ the hyperoctahedral group is a special case of a generalized symmetric group.

• Let p be a prime and let n≥1. Let P be a Sylow p-subgroup of the symmetric group S_pn of degree pn. Then P is isomorphic to the iterated regular wreath product W_n = ℤ_p ≀ ℤ_p≀...≀ℤ_p of n copies of ℤ_p. Here W₁ := ℤ_p and W_k := W_k-1≀ℤ_p for all k≥2.

• The Rubik's Cube group is a subgroup of small index in the product of wreath products, (ℤ₃≀S₈) × (ℤ₂≀S₁₂), the factors corresponding to the symmetries of the 8 corners and 12 edges.