Examples
- The Lamplighter group is the restricted wreath product ℤ2≀ℤ.
- ℤm≀Sn (Generalized symmetric group).
- The base of this wreath product is the n-fold direct product
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- ℤmn = ℤm × ... × ℤm
- of copies of ℤm where the action φ : Sn → Aut(ℤmn) of the symmetric group Sn of degree n is given by
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- φ(σ)(α1,..., αn) := (ασ(1),..., ασ(n)).
- S2≀Sn (Hyperoctahedral group).
- The action of Sn 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 Spn of degree pn. Then P is isomorphic to the iterated regular wreath product Wn = ℤp ≀ ℤp≀...≀ℤp of n copies of ℤp. Here W1 := ℤp and Wk := Wk-1≀ℤp 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.
Read more about this topic: Wreath Product
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