Element structure of projective semilinear group of degree two over a finite field

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This article gives specific information, namely, element structure, about a family of groups, namely: projective semilinear group of degree two.
[[:Category:element structure {{{connective}}} group families|View element structure {{{connective}}} group families]] | View other specific information about projective semilinear group of degree two
[[Category:element structure {{{connective}}} group families]]

This article discusses the element structure of the projective semilinear group of degree two over a finite field.

We denote the field size by q and the field characteristic by p. We define r = \log_p q, so the field \mathbb{F}_q is a Galois extension of \mathbb{F}_p of degree r with cyclic Galois group.

Summary

Item Value
order of the group r(q^3 - q)
number of conjugacy classes For p \ne 2, expressible as a polynomial of degree r in p (see section #Number of conjugacy classes)

Conjugacy class structure

Number of conjugacy classes

For p \ne 2, the number of conjugacy classes can be expressed as a polynomial of degree r in p, where the polynomial depends only on r and not on p. Below are the first few cases:

Value of r Number of conjugacy classes case p = 2 Number of conjugacy classes case odd p (is a polynomial in p of degree r) Additional comments
1 3 p + 2 Note that in this case the group coincides with the projective general linear group of degree two, so the result follows from element structure of projective general linear group of degree two over a finite field
2 7 (p^2 + 3p + 8)/2