论文标题
关于群体的表演，单态和同构的数量
On the number of epi-, mono-, and homomorphisms of groups
论文作者
论文摘要
众所周知，从组$ f $到$ g $的同构数量是$ g $的最大共同分数和$ f/[f，f] $的指数。我们研究了满足某些自然条件（例如注射率或过渡性）的同态数量。我们结果的最简单的非平地推论是以下事实：{\ IT在任何有限的组中，生成对$（x，y）$的生成$ x^3 = 1 = y^5 $，是15的最大常见分裂的倍数，是组$ $ $ $ $ $ x^g，g，g，g] \ cdot \ cdot \ cdot \ cdot \ c^g^{g^{15} $。
It is known that the number of homomorphisms from a group $F$ to a group $G$ is divisible by the greatest common divisor of the order of $G$ and the exponent of $F/[F,F]$. We investigate the number of homomorphisms satisfying some natural conditions such as injectivity or surjectivity. The simplest nontrivial corollary of our results is the following fact: {\it in any finite group, the number of generating pairs $(x,y)$ such that $x^3=1=y^5$, is a multiple of the greatest common divisor of 15 and the order of the group $[G,G]\cdot\{g^{15}\;|\;g\in G\}$.
