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Determining sets and determining numbers of finite groups. (arXiv:1801.08456v1 [math.GR])
来源于:arXiv
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 th 查看全文>>