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This paper continues previous work, based on systematic use of a formula of L. Scott, to detect Hurwitz groups. It closes the problem of determining the finite simple groups contained in $PGL_n(F)$ for $nleq 7$ which are Hurwitz, where $F$ is an algebraically closed field. For the groups $G_2(q)$, $qgeq 5$, and the Janko groups $J_1$ and $J_2$ it provides explicit $(2,3,7)$-generators.
We classify the finitely generated prosupersolvable groups that satisfy Schreiers formula for the number of generators of open subgroups.
We present a number of examples to illustrate the use of small quotient dessins as substitutes for their often much larger and more complicated Galois (minimal regular) covers. In doing so we employ several useful group-theoretic techniques, such as
Let $M$ be a compact surface without boundary, and $ngeq 2$. We analyse the quotient group $B_n(M)/Gamma_2(P_n(M))$ of the surface braid group $B_{n}(M)$ by the commutator subgroup $Gamma_2(P_n(M))$ of the pure braid group $P_{n}(M)$. If $M$ is diffe
Divergence functions of a metric space estimate the length of a path connecting two points $A$, $B$ at distance $le n$ avoiding a large enough ball around a third point $C$. We characterize groups with non-linear divergence functions as groups having
We prove that all invariant random subgroups of the lamplighter group $L$ are co-sofic. It follows that $L$ is permutation stable, providing an example of an infinitely presented such a group. Our proof applies more generally to all permutational wre