arXiv subjects
Publications and source records attributed to Kaiwen Lu.
Let $X/K$ be an abelian variety defined over a number field and let $f:X\to X$ be a dominant morphism defined over $K$. We show that $(f,K)$ is strongly dense orbit transversal. That is, up to replacing $K$ with a finite extension, every set of representatives for grand $(f,K)$-orbits is Zariski dense in $X$.
Let $X/K$ be a smooth projective variety defined over a number field and $f:X\to X$ be a morphism defined over $K$. Assuming there exists a point in $X(K)$ whose $f$-orbit is Zariski dense in $X$ and up to replacing $K$ by a finite extension, Pasten and Silverman studied the distribution of grand $(f,K)$-orbits and proved that many sets of representatives of grand $(f,K)$-orbits on various classes of varieties are Zariski dense. In particular, they showed that if $X$ is a geometrically simple abelian variety, then all such sets of representatives are Zariski dense. We demonstrate the existence of a dense set of representatives for maps on all abelian varieties.
In this paper, we investigate finite solvable tidy groups. We classify the tidy $\{ p, q \}$-groups. Combining this with a previous result, we are able to characterize the finite tidy solvable groups. Using this characterization, we bound the Fitting height of finite tidy solvable groups and we prove that the quotients of finite tidy solvable groups are tidy.
Morgan and Parker have proved that if $G$ is a group satisfying the condition that $Z(G) = 1$, then the connected components of the commuting graph of $G$ have diameter at most $10$. Parker has proved that if in addition $G$ is solvable, then the commuting graph of $G$ is disconnected if and only if $G$ is a Frobenius group or a $2$-Frobenius group, and if the commuting graph of $G$ is connected, then its diameter is at most $8$. We prove that the hypothesis $Z (G) = 1$ in these results can be replaced with $G' \cap Z(G) = 1$. We also prove that if $G$ is solvable and $G/Z(G)$ is either a Frobenius group or a $2$-Frobenius group, then the commuting graph of $G$ is disconnected.