Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Determining sets and determining numbers of finite groups   Let $G$ be a group. A subset $D$ of $G$ is a determining set of $G$, if every
automorphism of $G$ is uniquely determined by its action on $D$. The
determining number of $G$, denoted by $\alpha(G)$, is the cardinality of a
smallest determining set. A generating set of $G$ is a subset such that every
element of $G$ can be expressed as the combination, under the group operation,
of finitely many elements of the subset and their inverses. The cardinality of
a smallest generating set of $G$, denoted by $\gamma(G)$, is called the
generating number of $G$. A group $G$ is called a DEG-group if
$\alpha(G)=\gamma(G)$.
  The main results of this article are as follows. Finite groups with
determining number $0$ or $1$ are classified; Finite simple groups and finite
nilpotent groups are proved to be DEG-groups; A finite group is a normal
subgroup of a DEG-group and there is an injective mapping from the set all
finite groups to the set of finite DEG-groups; Nilpotent groups of order $n$
which have the maximum determining number are classified; For any integer
$k\geq 2$, there exists a group $G$ such that $\alpha(G)=2$ and $\gamma(G)\geq
k$.

5
The categories of this paper are: gr-qc
4
The categories of this paper are: cond-mat
12
The categories of this paper are: nucl-th
9
The categories of this paper are: nlin
22
The categories of this paper are: chao-dyn
0
The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci
37
The categories of this paper are: supr-con
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
18
The categories of this paper are: econ
33
The categories of this paper are: plasm-ph
3
The categories of this paper are: physics