Does Cp rwr Cp have more than two isomrphic class?
It is well known that the Sylow-p-subgroup of the Symmetric group S(p^k) [where p is a prime & k is a positive integer] is the iterated regular wreath product (with k copies of Cp)-
(...((Cp rwr Cp)rwr Cp)...rwr Cp). Now, this group must point toward one isomorphic class because Sylow-p-subgroups are conjugate & hence isomorphic. But, according to the definition of regular wreath product G rwr H, one needs to specify a sequence of the elements of the group H. Because each sequence (ordering) of the elements gives rise to different auto-morphisms. But, here in the case of the Sylow-p-subgroup of S(p^k), no such ordering is specified. Is it so because any ordering of the H at hand (Cp in this case) will point toward one isomorphic class or should I, by default, assume that the ordering is- 1, x, x^2, x^3,....,x^p where x is the generator of Cp ?
I think that the ordering does not matter here or else the book which I am following would have specified out the ordering. (I am following 'A course in group theory' by John F. Humphreys page 167.) But I cannot prove this fact- Why would G rwr Cp would point out one isomorphic class for all ordering of the elements of Cp?
I will probably understand every term you write like semi-direct products or actions etc. So, please answer or give me a useful link.