Elementary abelian subgroups of prime-square order in unitriangular matrix group:UT(3,p)

From Groupprops
Jump to: navigation, search
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.
VIEW: Group-subgroup pairs with the same subgroup part | Group-subgroup pairs with the same group part| Group-subgroup pairs with the same quotient part | All pages on prime-parametrized particular subgroups in groups

Let p be an odd prime (The case p = 2, where we get Klein four-subgroups of dihedral group:D8 and cyclic maximal subgroup of dihedral group:D8, is somewhat different, though many things are similar).

The group G = U(3,p) is the group of unipotent upper-triangular matrices over the prime field of p elements, and is also the unique (up to isomorphism) non-abelian group of order p^3 and exponent p. It is defined by the presentation:

G := \langle g,h,k \mid g^p = h^p = k^p = e, gh = hg, gk = kg, khk^{-1} = g^{-1}h \rangle.

In the matrix description, each matrix (a_{ij}) can be described by the three entries a_{12}, a_{13}, a_{23}. The matrix looks like:

\begin{pmatrix}
1 & a_{12} & a_{13} \\
0 & 1 & a_{23}\\
0 & 0 & 1\end{pmatrix}

The multiplication of matrices A = (a_{ij}) and B = (b_{ij}) gives the matrix C = (c_{ij}) where:

  • c_{12} = a_{12} + b_{12}
  • c_{13} = a_{13} + b_{13} + a_{12}b_{23}
  • c_{23} = a_{23} + b_{23}

With the matrix description, we can set g as the matrix with a_{13} = 1 and the other two entries zero, h as the matrix with a_{12} = 1 and the other two entries zero, and k as the matrix with a_{23} = 1 and the other two entries zero.

Now define the subgroup:

H_0 := \{ \begin{pmatrix}
1 & a_{12} & a_{13} \\
0 & 1 & 0\\
0 & 0 & 1\end{pmatrix} : a_{12}, a_{13} \in \mathbb{F}_p \}

In addition to this, consider the following p subgroups, parametrized by \{ 1,2,3,\dots,p \}:

H_i := \{ \begin{pmatrix}
1 & ia_{23} & a_{13} \\
0 & 1 & a_{23}\\
0 & 0 & 1\end{pmatrix} : a_{13},a_{23} \in \mathbb{F}_p \}

All the p + 1 subgroups H_0, H_1, \dots, H_p 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 H_0 and H_p in particular are pattern subgroups with respect to the given choice of basis.

Arithmetic functions

Function Value Explanation
order of whole group p^3
order of subgroup p^2
index of subgroup p
size of conjugacy class of subgroup 1
number of conjugacy classes in automorphism class of subgroup p + 1
size of automorphism class of subgroup p + 1