This essay focuses on features of a geometrical.The discussion group questions are on the first page of the attache document.
Read both of the assigned papers. Answer the discussion group questions. Find two NEW papers on the same topic. And write a 2 page essay synthesizing all the materials (two assign papers, two NEW papers, and answering the discussion group questions). The discussion group questions are on the first page of the attache document.
For example, a symmetry group encodes symmetry features of a geometrical object. The group consists of the set of transformations that leave the object change. The operation of combining two such transformations by performing one after the other. Lie groups arise as symmetry groups in geometry but appear also in the Standard Model of particle physics. The Poincaré group is a Lie group consisting of the symmetries of spacetime in special relativity.
The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. Who introduce the term of group (groupe, in French) for the features. Geometrical symmetry group of the roots of an equation, now call a Galois group. After contributions from other fields such as number theory and geometry, the group notion was generalize and firmly establish around 1870.
To explore groups, mathematicians have devised various features of a geometrical notions to break groups into smaller. Better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be express concretely. Both from a point of view of representation theory (that is. Through the representations of the group) and of computational group theory.
A theory has been develope for finite groups, which culminate with the classification of finite simple groups, complete in 2004.Since the mid-1980s, geometric group theory. Which studies finitely generate groups as geometric objects, has become an active area in group theory.