Computational group theory and automorphisms of relatively free groups.

我的研究建议是寻求关于自同构的以下基本问题的解决方案。 (1)刻画自由可解群的自同构。这个问题是由于Mal‘cev，Kargapolov，Remeslennikov(Kourovka笔记本中未解决的问题)造成的。这个问题需要找到一个解：给定的n×n导数矩阵(n大于或等于3)在某个自由可解群环上是否可逆？ (2)素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-&gt；(W)Phi_1-&gt；(W)Phi_1Phi_2-&gt；...-&gt；(W)Phi_1Phi_2……Phi_k=x_1。在这里，我建议找出对某些相对自由的群，m元系在F/V中是本原的一个判据(例如，V=[F‘’，F‘’])。 (3)算法问题：群G的自同构可约性问题是可判定的，如果存在一个算法来决定对G的每一对元素g，h：在Aut(G)中是否存在自同构Phi，如(G)Phi=h？[这是对自由群体的著名的怀特黑德问题的类比(怀特黑德(1936)，拉帕波特(1952)，林登，希金斯(1974))。1997年，作为第一步，Timoshenko和我证明了对于一个秩为2的自由亚贝尔群，自同构可约性问题是可判定的。我建议对某些相对自由群找到上述性质成立的充要条件。 (4)算法问题：群G中的一个测试元是由G的一个自同态Phi确定的元素g，则Phi一定是自同构。我建议找到一种算法来确定：给定的单词w是否为某些相对自由的组中的测试元素；我还将寻找测试元素的描述。 接下来，我将继续研究以下几个主题： (I)部分交换亚贝尔群的普遍理论是可判定的吗？ (Ii)有限基问题：群的非特殊变种和群表示； (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)，其中n大于或等于2)； (Iv)找出两个一般2×2矩阵的代数的新的自同构。在(1997,2000,2005)中，Drensky和我构造了2×2一般矩阵的新的自同构。

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.