An affirmative answer to a conjecture related to the solvability of groups

More on a question of M. Newman on isomorphic subgroups of solvable groups

M.Newman has asked if it is the case that whenever H and K are isomorphic subgroups of a finite solvable group G with H maximal, then K is also maximal. This question was considered in a paper of I.M. Isaacs and the second author, where (among other things) the answer was shown to be affirmative if H has an Abelian Sylow 2-subgroup. Here, we show that the answer is affirmative unless the index of H is a power of a prime less than 5 and we obtain further restrictions on the structure of a purported minimal counterexample.

Normal subgroups of limit groups of prime index

Motivated by their study of pro-p limit groups, D.H. Kochloukova and P.A. Zalesskii formulated a question concerning the minimum number of generators d(N) of a normal subgroup N of prime index p in a non-abelian limit group G (cf. Question*). It is shown that the analogous question for the rational rank has an affirmative answer (cf. Thm. A). From this result one may conclude that the original question of D.H. Kochloukova and P.A. Zalesskii has an affirmative answer if the abelianization G^{\ab} of G is torsion free and d(G)=d(G^{\ab}) (cf. Cor.~B), or if G has the IF-property (cf. Thm. C).

Pronormality of Hall subgroups in finite simple groups

We prove that Hall subgroups of finite simple groups are pronormal. Thus we obtain an affirmative answer to Problem 17.45(a) of "Kourovka notebook".

无真异常子群的无限单群

Infinite simple groups with no proper abnormal subgroups

本文利用Obraztsov嵌入定理构造了无真异常子群的无限单群，肯定回答了Revin的问题，并否定回答了Kurdachenko等人的问题。

Intersection problem for Droms RAAGs

We solve the subgroup intersection problem (SIP) for any RAAG G of Droms type (i.e., with defining graph not containing induced squares or paths of length 3): there is an algorithm which, given finite sets of generators for two subgroups H,K of G, decides whether $H \cap K$ is finitely generated or not, and, in the affirmative case, it computes a set of generators for $H \cap K$. Taking advantage of the recursive characterization of Droms groups, the proof consists in separately showing that the solvability of SIP passes through free products, and through direct products with free-abelian grou…

Diamond Theorem for finitely generated Profinite free Group

We extend Haran's Diamond Theorem to closed subgroups of a finitely generated free profinite group. This gives an affirmative answer to Problem 25.4.9 in the book Field Arithmetics of Fried and Jarden.