Affiliation:
1. Department of Mathematics , Beijing Jiaotong University , Beijing , 100044 , P. R. China
2. UP IAM and UP FAMNIT , University of Primorska , Muzejski trg 2, SI-6000 Koper , Slovenia
Abstract
Abstract
In this paper, we characterize the finite groups 𝐺 of even order with the property that, for any involution 𝑥 and element 𝑦 of 𝐺,
⟨
x
,
y
⟩
\langle x,y\rangle
is isomorphic to one of the following groups:
Z
2
\mathbb{Z}_{2}
,
Z
2
2
\mathbb{Z}_{2}^{2}
,
Z
4
\mathbb{Z}_{4}
,
Z
6
\mathbb{Z}_{6}
,
Z
2
×
Z
4
\mathbb{Z}_{2}\times\mathbb{Z}_{4}
,
Z
2
×
Z
6
\mathbb{Z}_{2}\times\mathbb{Z}_{6}
and
A
4
A_{4}
.
As a result, a characterization will be obtained for the finite groups all of whose Cayley graphs of degree 3 have integral spectrum.
Subject
Algebra and Number Theory