论文标题
涂鸦组的生成图
The generating graph of a profinite group
论文作者
论文摘要
令$ g $为2生成组。 $ g $的生成图$γ（g）$是其顶点是$ g $的元素，如果$ g = \ langle g，h \ rangle。$可以扩展到2-生成的profinite $ g，$考虑在此情况中，请考虑将其扩展到该案例$ g，考虑在此情况中，请考虑将两个顶点$ g $和$ h $相邻。我们证明，$γ（g）$的非异形顶点的集合$ v（g）$以$ g $的关闭，如果$ g $是prosoluble的，则该图$δ（g）$从$γ（g）$获得的$γ（g）$取消，通过删除其隔离的顶点与Diameters compatient uptime $ gyt uppert $ gatient $ gatient。 $δ（g）$具有$ 2^{\ aleph_0} $连接的组件。这意味着所谓的“交换猜想”不适合有限生成的涂鸦组。我们还证明，如果$ v（g）$的元素在图$γ（g）中具有有限的学位，则$ g $是有限的。
Let $G$ be 2-generated group. The generating graph $Γ(G)$ of $G$ is the graph whose vertices are the elements of $G$ and where two vertices $g$ and $h$ are adjacent if $G = \langle g, h \rangle.$ This definition can be extended to a 2-generated profinite group $G,$ considering in this case topological generation. We prove that the set $V(G)$ of non-isolated vertices of $Γ(G)$ is closed in $G$ and that, if $G$ is prosoluble, then the graph $Δ(G)$ obtained from $Γ(G)$ by removing its isolated vertices is connected with diameter at most 3. However we construct an example of a 2-generated profinite group $G$ with the property that $Δ(G)$ has $2^{\aleph_0}$ connected components. This implies that the so called "swap conjecture" does not hold for finitely generated profinite groups. We also prove that if an element of $V(G)$ has finite degree in the graph $Γ(G),$ then $G$ is finite.
