Wreath Product - Notation and Conventions

Notation and Conventions

The structure of the wreath product of A by H depends on the H-set Ω and in case Ω is infinite it also depends on whether one uses the restricted or unrestricted wreath product. However, in literature the notation used may be deficient and one needs to pay attention on the circumstances.

• In literature A≀_ΩH may stand for the unrestricted wreath product A Wr_Ω H or the restricted wreath product A wr_Ω H.

• Similarly, A≀H may stand for the unrestricted regular wreath product A Wr H or the restricted regular wreath product A wr H.

• In literature the H-set Ω may be omitted from the notation even if Ω≠H.

• In the special case that H = S_n is the symmetric group of degree n it is common in the literature to assume that Ω={1,...,n} (with the natural action of S_n) and then omit Ω from the notation. That is, A≀S_n commonly denotes A≀_{1,...,n}S_n instead of the regular wreath product A≀_SnS_n. In the first case the base group is the product of n copies of A, in the latter it is the product of n! copies of A.