# Inner automorphism group of wreath product of groups of order p

## Definition

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.

## Particular cases

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 |

## Families

Larger collections of groups in which this group figures: