Structure of a Conjugating Automorphism Group

We examine the automorphism group Aut(F _n ) of a free group F _n of rank n 2 on free generators x ₁, x ₂,...,x _n . It is known that Aut(F ₂) can be built from cyclic subgroups using a free and semidirect product. A question remains open as to whether this result can be extended to the case n > 2. Every automorphism of Aut(F _n ) sending a generator x _i to an element f _i ^-1 x _(i) f _i , where f _i F _n and is some permutation on a symmetric group S _n , is called a conjugating automorphism. The conjugating automorphism group is denoted C _n . A set of automorphisms for which is the identity permutation form a basis-conjugating automorphism group, denoted Cb _n . It is proved that Cb _n can be factored into a semidirect product of some groups. As a consequence we obtain a normal form for words in C _n . For n 4, C _n and Cb _n have an undecidable occurrence problem in finitely generated subgroups. It is also shown that C _n , n 2, is generated by at most four elements, and we find its respective genetic code, and that Cb _n , n 2, has no proper verbal subgroups of finite width.     

关于G逆半群的同余格

本文讨论Ｇ逆半群上的同余,文中给出刻划ｐ＾ｋ,ｐｋ,ｐ＾Ｔ,ｐＴ的方法,并用来讨论半格同余,纯幂同余,群同余及幂等元分离同余。     

非平凡正规子群的阶相同的有限群

本文研究了非平凡正规子群的阶相同的有限群. 给出了非平凡正规子群唯一的可解群的结构. 对非可解群的情况, 我们也给出了它的刻画.     

Transitive Permutation Groups with Cyclic Point Stabilizers of Maximum Order

We prove that a transitive permutation group of degree n with a cyclic point stabilizer and whose order is n(n-1) is isomorphic to the affine group of degree 1 over a field with n elements. More generally we show that if a finite group G has an abelian and core-free Hall subgroup Q, then either Q has a small order (2|Q|² < |G|) or G is a direct product of 2-transitive Frobenius groups.     

On Atkin-Swinnerton-Dyer congruence relations

In this paper, we exhibit a noncongruence subgroup Γ whose space of weight 3 cusp forms S₃(Γ) admits a basis satisfying the Atkin-Swinnerton-Dyer congruence relations with two weight 3 newforms for certain congruence subgroups. This gives a modularity interpretation of the motive attached to S₃(Γ) by Scholl and also verifies the Atkin-Swinnerton-Dyer congruence conjecture for this space.     

A conjecture of Abe-Iiyori and the Ree groups ~2F_4(q),Ⅱ

In this paper the author has solved a problem of Abe and liyori for the finite simple groups 2F4(q) and 2F4(2)'.     

Groups whose cyclic subgroups have bounded permutability properties

It follows from classical results of Neumann and Macdonald that a group G has finite commuator subgroup if and only if either the normalizers of cyclic subgroups of G have boundedly finite indices or cyclic subgroups of G have bounded indices in their normal closures. In this paper, groups with a similar condition are considered, when normality is replaced by permutability.      

超可解群的若干充分条件

本文研究了有限群的超可解性问题，利用子群的共轭可换性概念及极小反例法，获得了一个群为超可解的若干充分条件．举例说明了主要结果中的假设条件是不可少的．