Relation To Efficient Presentations
In combinatorial group theory, a group often originates from a presentation. One important theme in this area of mathematics is to study presentations with as few relations as possible, such as one relator groups like Baumslag-Solitar groups. These groups are infinite groups with two generators and one relation, and an old result of Schreier shows that in any presentation with more generators than relations, the resulting group is infinite. The borderline case is thus quite interesting: finite groups with the same number of generators as relations are said to have a deficiency zero. For a group to have deficiency zero, the group must have a trivial Schur multiplier because the minimum number of generators of the Schur multiplier is always less than or equal to the difference between the number of relations and the number of generators, which is the negative deficiency. An efficient group is one where the Schur multiplier requires this number of generators.
A fairly recent topic of research is to find efficient presentations for all finite simple groups with trivial Schur multipliers. Such presentations are in some sense nice because they are usually short, but they are difficult to find and to work with because they are ill-suited to standard methods such as coset enumeration.
Read more about this topic: Schur Multiplier
Famous quotes containing the words relation to, relation and/or efficient:
“Science is the language of the temporal world; love is that of the spiritual world. Man, indeed, describes more than he explains; while the angelic spirit sees and understands. Science saddens man; love enraptures the angel; science is still seeking, love has found. Man judges of nature in relation to itself; the angelic spirit judges of it in relation to heaven. In short to the spirits everything speaks.”
—Honoré De Balzac (1799–1850)
“Parents ought, through their own behavior and the values by which they live, to provide direction for their children. But they need to rid themselves of the idea that there are surefire methods which, when well applied, will produce certain predictable results. Whatever we do with and for our children ought to flow from our understanding of and our feelings for the particular situation and the relation we wish to exist between us and our child.”
—Bruno Bettelheim (20th century)
“The really efficient laborer will be found not to crowd his day with work, but will saunter to his task surrounded by a wide halo of ease and leisure.”
—Henry David Thoreau (1817–1862)