arXiv subjects
Publications and source records attributed to Stefan-Christoph Virchow.
We use character theory and character estimates to show that the number of $T_2$-systems of $A_n$ is at least \begin{equation*} \frac{1}{8n\sqrt{3}}\exp\left(\frac{2π}{\sqrt{6}}n^{1/2}\right)(1+o(1)). \end{equation*} Applying this result, we obtain a lower bound for the number of connected components of the product replacement graph $Γ_2(A_n)$.
We consider the $q$th root number function for the symmetric group. Our aim is to develop an asymptotic formula for the multiplicities of the $q$th root number function as $q$ tends to $\infty$. We use character theory, number theory and combinatorics.
We consider the probability $p(S_n)$ that a pair of random permutations generates either the alternating group $A_n$ or the symmetric group $S_n$. Dixon (1969) proved that $p(S_n)$ approaches $1$ as $n\to\infty$ and conjectured that $p(S_n)=1-1/n+o(1/n)$. This conjecture was verified by Babai (1989), using the Classification of Finite Simple Groups. We give an elementary proof of this result; specifically we show that $p(S_n)=1-1/n+\mathcal {O}(n^{-2+ε})$. Our proof is based on character theory and character estimates, including recent work by Schlage-Puchta (2012).