Inner automorphism group of wreath product of groups of order p

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Definition

Let p be a prime number. The inner automorphism group of wreath product of groups of order p is a group obtained as follows: it is the inner automorphism group of the wreath product of groups of order p, where the acting group acts regularly: it acts by cyclic permutation of p coordinates.

In other words, if \mathbb{Z}_p denotes the cyclic group of order p, and G = \mathbb{Z}_p \wr \mathbb{Z}_p = (\mathbb{Z}_p \times \dots \times \mathbb{Z}_p) \rtimes \mathbb{Z}_p, then the group we want is H = G/Z(G). It turns out that Z(G) is precisely the diagonal subgroup of \mathbb{Z}_p \times \dots \times \mathbb{Z}_p.

Thus, the group H that we want is a group of order p^p. It turns out that for p odd, this group is a maximal class group (for p = 2, it is isomorphic to the Klein four-group). For all p, it has exponent p. Since its nilpotence class is p - 1, it is a regular p-group; however, since it has exponent p and size p^p, it is not an absolutely regular p-group.

Particular cases

Prime number p Wreath product \mathbb{Z}_p \wr \mathbb{Z}_p Group of interest Order p^p Nilpotency class p - 1
2 dihedral group:D8 Klein four-group 4 1
3 wreath product of Z3 and Z3 unitriangular matrix group:UT(3,3) 27 2
5 wreath product of Z5 and Z5 inner automorphism group of wreath product of Z5 and Z5 3125 4

Families

Larger collections of groups in which this group figures: