Elementary abelian subgroups of prime-square order in unitriangular matrix group:UT(3,p)
This article is about a particular subgroup in a group parametrized by a prime (so we get one group for each prime and a corresponding subgroup for it), up to equivalence of subgroups (i.e., an isomorphism of groups that induces the corresponding isomorphism of subgroups). The subgroup is (generically) elementary abelian group of prime-square order and the group is (generically) prime-cube order group:U(3,p) (see subgroup structure of prime-cube order group:U(3,p)).
The subgroup is a normal subgroup and the quotient group is (generically) group of prime order.
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The group is the group of unipotent upper-triangular matrices over the prime field of elements, and is also the unique (up to isomorphism) non-abelian group of order and exponent . It is defined by the presentation:
In the matrix description, each matrix can be described by the three entries . The matrix looks like:
The multiplication of matrices and gives the matrix where:
With the matrix description, we can set as the matrix with and the other two entries zero, as the matrix with and the other two entries zero, and as the matrix with and the other two entries zero.
Now define the subgroup:
In addition to this, consider the following subgroups, parametrized by :
All the subgroups are elementary abelian subgroups of prime-square order and they are all automorphic subgroups, i.e., they are all related by automorphisms of the whole group.
Note that and in particular are pattern subgroups with respect to the given choice of basis.
|order of whole group|
|order of subgroup|
|index of subgroup|
|size of conjugacy class of subgroup||1|
|number of conjugacy classes in automorphism class of subgroup|
|size of automorphism class of subgroup|