# Changes

## Unitriangular matrix group:UT(3,p)

, 16:43, 10 September 2012
Subgroups
==Subgroups==
{{further|[[Subgroup structure of prime-cube order unitriangular matrix group:UUT(3,p)]]}}
Here is the complete list of subgroups{{#lst# The trivial subgroup (1)# The center, which is a group structure of order [itex]p[/itex]. In unitriangular matrix terms, this is the subgroup comprising matrices [itex]a_{ij}[/itex] with [itex]a_{12} = a_{23} = 0[/itex]. group:UT(1)# Subgroups of order [itex]p[/itex] generated by non-central elements. These are not normal3, and occur in conjugacy classes of size [itex]p[/itex]. ([itex]p(p+1)[/itex])# Subgroups of order [itex]p^2[/itex] containing the center. These are the inverse images via the quotient map by the center, of subgroups of order [itex]p[/itex] in the [[inner automorphism group]]. ([itex]p + 1[/itex])# The whole group. (1) {{normal subgroups}} The subgroups in (1), (2), (4) and (5) above are normal. {{characteristic subgroups|summary}} The subgroups in (1), (2) and (5) above are normal. In other words, there are only three characteristic subgroups. Some notable facts: * The group is [[characteristic-comparable group|characteristic-comparable]]: any two characteristic subgroups can be compared* More generally, any characteristic subgroup and any normal subgroup can be compared.* The characteristic subgroups are precisely the subgroups that occur in the [[derived series]], [[upper central series]] and [[lower central series]].
==Linear representation theory==