Authors
Shilin I. A.
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Degree
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Ph. D. (Math.), Associate Professor, Higher Mathematics Department, M. Sholokhov Moscow State University for the Humanities, Mathematical Modeling Department, Moscow Aviation Institute |
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E-mail
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ilyashilin@li.ru |
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Location
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Moscow |
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Articles
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Homomorphism groups computing and homomorphic stability of pairs of finite groups verificationWe consider all pairs of groups G and H under the condition that their orders are not greater than 12. For Abelian group H, we obtain (up to isomorphism) via computer modeling, first, the group of homomorphisms of G to H and, second, the subgroup in H which is a union of the images of all homomorphisms. For non-Abelian H, we consider the similar constrictions, but this case is more difficult. Read more...Programming method for computing topologies and study their propertiesWe suggest the algorithm and the corresponding programming code for computing topologies of a finite set. Algorithm provides verification of T0-axiom of the separation and connectivity, computing the connected components, bases and topological spaces homeomorphisms to itself. Software implementing the proposed model developed by the authors is presented.
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One the computing complex for solving problems related to finite groupsIn this article we describe the special computing program created at The Center for Research and
Education in Mathematics of Sholokhov Moscow State University for the Humanities. This program is
created for university students learning an important part of abstract algebra course known as group
theory. It is well known that teaching and learning group theory have a lot of difficulties related to very
abstract level of group theory concepts. It means, in particular, that students need any methods to
visualize these concepts. Nowadays universities have good software for group theoretical researches
(for instance, GAP, Magma, Cayley, etc.), but all these computer algebra systems are very useful
for namely for researches and almost useless for students because they want to see how to solve a
problem step by step. The main goal of our program is to help students to take part in solving of the
most typical problems arising in finite group theory. By using this program, the student can find all
subgroups of a given group, separate all normal divisors, obtain the normalizer and centralizer for
a subgroup, compute left and right cosets with respect to a given normal divisor, represent a given
group as a group of permutations (to embed a group to appropriate symmetric group), represent a
permutation as a product of independent cycles and a composition of transposition (of general and
special forms), compute the order and power of a permutation. This program contains the library of
more than 50 finite groups. It is easy to add new options to our program for example, (to compute
the automorphism and homomorphism groups for a given group of pair of groups, where the second
group is Abelian).
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