Computational group theory and automorphisms of relatively free groups.

我的研究计划是寻求以下关于自同构的基本问题的解决方案。 (1)描述自由可解群的自同构。这个问题是由于马尔采夫，Kargapolov，Remeslennikov（未解决的问题在Kourovka笔记本）。这个问题将需要找到一个解决方案：是否一个给定的n × n导子矩阵（n大于或等于3）是可逆的某些自由可解群环？ (2)本原性是自同构的另一个方面。F中的一个元素是本原的，如果它可以包含在F的一个基中。我们的主要问题是：给定一个词w，我们能否有效地构造一个将w映射到x_1的自同构序列，即，w->（w）phi_1->（w）phi_2->. - >（w）phi_1 phi_2. phi_k = x_1。在这里，我建议找到一个标准，一个系统的m个元素是原始的F/V（例如，V =[F "，F"]）的某些相对自由的群体。 (3)数学问题：群G的自同构约化问题是可判定的，如果存在一个算法，该算法判定以下问题：对于G的每一对元素g，h：Aut（G）中是否存在自同构φ，使得（g）φ = h？[This是自由群的著名的Whitehead问题的类似物（Whitehead（1936），Rappaport（1952），Lyndon，Higgens（1974））。在1997年，作为第一步，我证明了自守约化问题对于秩为2的自由亚阿贝尔群是可判定的。我建议找到必要和充分的条件，使上述性质对某些相对自由的群体成立。 (4)数学问题：群G中的一个测试元是由G的一个自同态φ固定的一个元g，则φ必是一个自同构。我建议找到一个算法来决定：一个给定的词w是否是某些相对自由的组中的测试元素;我还将寻求测试元素的描述。 接下来，我将继续研究以下主题： (i)部分交换亚交换群的泛理论是可判定的吗？ (ii)有限基问题：群和群表示的非Specht簇; (iii)某些n × n三角矩阵的识别问题，n大于或等于3（类似于著名的Magnus嵌入（1932），n大于或等于2）; (iv)找到两个一般2 x 2矩阵代数的新自同构。在（1997，2000，2005）中，Drensky和我构造了2 × 2一般矩阵的新自同构。

MY RESEARCH PROPOSAL IS TO SEEK SOLUTIONS to the following fundamental questions about automorphisms. (1) Describe automorphisms of free solvable groups. This question is due to Mal'cev, Kargapolov, Remeslennikov (unsolved problems in The Kourovka Note Book). This question will require finding a solution: whether or not a given n x n derivation matrix (n greater than or equal to 3) is invertible over certain free solvable group ring? (2) Primitivity is another aspect of automorphisms. An element in F is primitive if it can be included in a basis of F. Our main question is: given a word w, can we effectively construct a sequence of automorphisms which will map w to x_1, that is, w -> (w)phi_1 -> (w)phi_1 phi_2 -> ... -> (w)phi_1 phi_2 ... phi_k = x_1. Here, I propose to find a criterion for a system of m elements to be primitive in F/V (for example, V = [F'',F'']) for certain relatively free groups. (3) Algorithmic problem: The problem of automorphic reducibility is decidable for a group G if there exists an algorithm which decides the following question that for each pair of elements g,h of G: is there an automorphism phi in Aut(G) such as (g)phi = h? [This is an analogue of the famous Whitehead problem for free groups (Whitehead (1936), Rappaport (1952), Lyndon, Higgens (1974)). In 1997, as a first step Timoshenko and I proved that the problem of automorphic reducibility is decidable for a free metabelian group of rank 2. I propose to find necessary and sufficient conditions for which the above property holds for certain relatively free groups. (4) Algorithmic problem: A test element in a group G is an element g which is fixed by an endomorphism phi of G, then phi must be an automorphism. I propose to find an algorithm to decide: whether or not a given word w is a test element in certain relatively free groups; I will also seek description of test elements. NEXT I WILL CONTINUE MY RESEARCH ON THE FOLLOWING TOPICS: (i) Is the universal theory of partially commutative metabelian groups decidable?; (ii) finite basis problem: just non-Specht varieties of groups and group representations; (iii) identification problem of certain n x n triangular matrices, n greater than or equal to 3 (analogous to a well-known Magnus embedding (1932) for n greater than or equal to 2); (iv) find new automorphisms of the algebra of two generic 2 x 2 matrices. In (1997, 2000, 2005), Drensky and I constructed new automorphisms of 2 x 2 generic matrices.