The conjugacy class structure of the Qd(p) groups

Colin McKenzie L; Jane H. Long

The conjugacy class structure of the Qd(p) groups

Colin McKenzie L
, Jane H. Long

Stephen F Austin State University

Research output: Other contribution

Abstract

In this paper, we provide a classification of the conjugacy classes in the special affine group Qd(p) = (Z_p x Z_p) \semidirect SL(2,p) for odd primes. We use the known conjugacy class structure in SL(2,p) to lift elements to their conjugacy class in Qd(p). More specifically, for an element in Qd(p), its conjugacy class depends on whether the matrix component of the element is the identity, has eigenvalue 1, which is split into two classes in SL(2,p), or whether it lies in some other conjugacy class in SL(2,p). In addition, we provide formulas, in terms of p, for both the number and sizes of conjugacy classes in Qd(p).

Original language	Undefined/Unknown
State	Published - Jul 1 2026

Publication series

Name	Faculty Publications

Access to Document

http://scholarworks.sfasu.edu/mathandstats_facultypubs/33

Cite this

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title = "The conjugacy class structure of the Qd(p) groups",

abstract = "In this paper, we provide a classification of the conjugacy classes in the special affine group Qd(p) = (Zp x Zp) \textbackslash{}semidirect SL(2,p) for odd primes. We use the known conjugacy class structure in SL(2,p) to lift elements to their conjugacy class in Qd(p). More specifically, for an element in Qd(p), its conjugacy class depends on whether the matrix component of the element is the identity, has eigenvalue 1, which is split into two classes in SL(2,p), or whether it lies in some other conjugacy class in SL(2,p). In addition, we provide formulas, in terms of p, for both the number and sizes of conjugacy classes in Qd(p).",

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T1 - The conjugacy class structure of the Qd(p) groups

AU - L, Colin McKenzie

AU - Long, Jane H.

PY - 2026/7/1

Y1 - 2026/7/1

N2 - In this paper, we provide a classification of the conjugacy classes in the special affine group Qd(p) = (Zp x Zp) \semidirect SL(2,p) for odd primes. We use the known conjugacy class structure in SL(2,p) to lift elements to their conjugacy class in Qd(p). More specifically, for an element in Qd(p), its conjugacy class depends on whether the matrix component of the element is the identity, has eigenvalue 1, which is split into two classes in SL(2,p), or whether it lies in some other conjugacy class in SL(2,p). In addition, we provide formulas, in terms of p, for both the number and sizes of conjugacy classes in Qd(p).

AB - In this paper, we provide a classification of the conjugacy classes in the special affine group Qd(p) = (Zp x Zp) \semidirect SL(2,p) for odd primes. We use the known conjugacy class structure in SL(2,p) to lift elements to their conjugacy class in Qd(p). More specifically, for an element in Qd(p), its conjugacy class depends on whether the matrix component of the element is the identity, has eigenvalue 1, which is split into two classes in SL(2,p), or whether it lies in some other conjugacy class in SL(2,p). In addition, we provide formulas, in terms of p, for both the number and sizes of conjugacy classes in Qd(p).

M3 - Other contribution

T3 - Faculty Publications

ER -