Q:Information: - Mathematics (from Greek "mathma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. - In group theory, Tietze transformations are used to transform a given presentation of a group into another, often simpler presentation of the same group. These transformations are named after Heinrich Franz Friedrich Tietze who introduced them in a paper in 1908. - A mathematical problem is a problem that is amenable to being represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more abstract nature, such as Hilbert's problems.<br> It can also be a problem referring to the nature of mathematics itself, such as Russell's Paradox. - In the mathematical field of graph theory, a cubic graph is a graph in which all vertices have degree three. In other words, a cubic graph is a 3-regular graph. Cubic graphs are also called trivalent graphs. - In topology, the Tietze extension theorem (also known as the TietzeUrysohnBrouwer extension theorem) states that continuous functions on a closed subset of a normal topological space can be extended to the entire space, preserving boundedness if necessary. - In mathematics graph theory is the study of "graphs", which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of "vertices", "nodes", or "points" which are connected by "edges", "arcs", or "lines". A graph may be "undirected", meaning that there is no distinction between the two vertices associated with each edge, or its edges may be "directed" from one vertex to another; see Graph (discrete mathematics) for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics. - In abstract algebra, the group isomorphism problem is the decision problem of determining whether two given finite group presentations present isomorphic groups. - In mathematics, one method of defining a group is by a presentation. One specifies a set "S" of generators so that every element of the group can be written as a product of powers of some of these generators, and a set "R" of relations among those generators. We then say "G" has presentation - In mathematics and abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. - In mathematics, a theorem is a statement that has been proved on the basis of previously established statements, such as other theoremsand generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally "deductive", in contrast to the notion of a scientific law, which is "experimental". - In the mathematical field of graph theory, Tietze's graph is an undirected cubic graph with 12 vertices and 18 edges. It is named after Heinrich Franz Friedrich Tietze, who showed in 1910 that the Mobius strip can be subdivided into six regions that all touch each other three along the boundary of the strip and three along its center line and therefore that graphs that are embedded onto the Mobius strip may require six colors. The boundary segments of the regions of Tietze's subdivision (including the segments along the boundary of the Mobius strip itself) form an embedding of Tietze's graph. - A mathematician is someone who uses an extensive knowledge of mathematics in his/her work, typically to solve mathematical problems. - In computability theory and computational complexity theory, a decision problem is a question in some formal system that can be posed as a yes-no question, dependant on the input values. Decision problems typically appear in mathematical questions of decidability, that is, the question of the existence of an effective method to determine the existence of some object or its membership in a set; some of the most important problems in mathematics are undecidable. - In mathematics, topology (from the Greek , "place", and , "study") is concerned with the properties of space that are preserved under continuous deformations, such as stretching and bending, but not tearing or gluing. This can be studied by considering a collection of subsets, called open sets, that satisfy certain properties, turning the given set into what is known as a topological space. Important topological properties include connectedness and compactness. - In mathematics, the four color theorem, or the four color map theorem, states that, given any separation of a plane into contiguous regions, producing a figure called a "map", no more than four colors are required to color the regions of the map so that no two adjacent regions have the same color. Two regions are called "adjacent" if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. - In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term "abstract algebra" was coined in the early 20th century to distinguish this area of study from the other parts of algebra. - Heinrich Franz Friedrich Tietze ( August 31 , 1880 -- February 17 , 1964 ) was an Austrian mathematician , famous for the Tietze extension theorem on functions from topological spaces to the real numbers . He also developed the Tietze transformations for group presentations , and was the first to pose the group isomorphism problem . Tietze 's graph is also named after him ; it describes the boundaries of a subdivision of the Mobius strip into six mutually - adjacent regions , found by Tietze as part of an extension of the four color theorem to non-orientable surfaces . What is the relationship between 'heinrich franz friedrich tietze' and 'group theory'?
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field of work