Fixed Points of Automorphisms of Certain Non-Cyclic p-Groups and the Dihedral Group

Fixed Points of Automorphisms of Certain Non-Cyclic p-Groups and the Dihedral Group

Let G=Zp⊕Zp2, where p is a prime number. Suppose that d is a divisor of the order of G. In this paper, we find the number of automorphisms of G fixing d elements of G and denote it by θ(G,d). As a consequence, we prove a conjecture of Checco-Darling-Longfield-Wisdom. We also find...

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Bibliographic Details
Main Authors: Umar Hayat, Daniel López-Aguayo, Akhtar Abbas
Format: Article
Language:English
Published: MDPI AG 2018-06-01
Series:Symmetry
Subjects:
fixed-point-free automorphism
dihedral group
theta values
unitriangular
Heisenberg group
Online Access:http://www.mdpi.com/2073-8994/10/7/238
Description
Summary:Let G=Zp⊕Zp2, where p is a prime number. Suppose that d is a divisor of the order of G. In this paper, we find the number of automorphisms of G fixing d elements of G and denote it by θ(G,d). As a consequence, we prove a conjecture of Checco-Darling-Longfield-Wisdom. We also find the exact number of fixed-point-free automorphisms of the group Zpa⊕Zpb, where a and b are positive integers with a<b. Finally, we compute θ(D2q,d), where D2q is the dihedral group of order 2q, q is an odd prime, and d∈{1,q,2q}.
ISSN:2073-8994

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