Inner automorphism group of wreath product of groups of order p
Let 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 coordinates.
In other words, if denotes the cyclic group of order , and , then the group we want is . It turns out that is precisely the diagonal subgroup of .
Thus, the group that we want is a group of order . It turns out that for odd, this group is a maximal class group (for , it is isomorphic to the Klein four-group). For all , it has exponent p. Since its nilpotence class is , it is a regular p-group; however, since it has exponent and size , it is not an absolutely regular p-group.
|Prime number||Wreath product||Group of interest||Order||Nilpotency class|
|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|
Larger collections of groups in which this group figures: