Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On finite groups with prescribed two-generator subgroups and integral
  Cayley graphs   In this paper, we characterize the finite groups $G$ of even order with the
property that for any involution $x$ and element $y$ of $G$, $\langle x, y
\rangle$ is isomorphic to one of the following groups: $\mathbb{Z}_2,$
$\mathbb{Z}_2^2$, $\mathbb{Z}_4$, $\mathbb{Z}_6$, $\mathbb{Z}_2 \times
\mathbb{Z}_4$, $\mathbb{Z}_2 \times\mathbb{Z}_6$ and $A_4$. As a result, a
characterization will be obtained for the finite groups all of whose Cayley
graphs of degree $3$ have integral spectrum.

4
The categories of this paper are: cond-mat
29
The categories of this paper are: funct-an
27
The categories of this paper are: comp-gas
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
23
The categories of this paper are: solv-int
33
The categories of this paper are: plasm-ph
28
The categories of this paper are: alg-geom
25
The categories of this paper are: patt-sol
36
The categories of this paper are: cmp-lg
26
The categories of this paper are: bayes-an
21
The categories of this paper are: adap-org
37
The categories of this paper are: supr-con
31
The categories of this paper are: ao-sci
13
The categories of this paper are: quant-ph
16
The categories of this paper are: stat