Groups arise in many mathematical settings and in studies invoving symmetries in physics and chemistry. For a concrete example of a group, consider the symmetries of the square. Since any combination of these motions is again one of them, they form a group. Below a multicolored square is pictured with the effects of these motions on it. 180 R 270 V H F 1 F 2 Research in group theory at the University of Florida is carried out by mathematicians in Algebra Research Group .
Groups arise in a wide variety of settings in different ways.
The Inverse Galois Problem concerns the question whether for any given group there exists a polynomial equation such that the Galois group associated with the solution of this equation is the given group. The conjecture is that is always the case, but the problem in full generality is still unsolved. Thompson has collaborated with mathematicians at the University of Florida on research related to the inverse Galois problem. Similarly, in group theory, one studies simple groups , those which have no normal subgroups except the identity and the whole group. For many years, the classification of all the finite simple groups was the most important problem in group theory.