Divide between group theory and mathematics

This article is about the growing divide between group theory and the rest of mathematics.

Groups are ubiquitous in mathematics: they occur in algebraic topology, differential geometry/topology, algebraic geometry, Lie theory, functional analysis, probability theory, number theory, combinatorics and practically all parts of mathematics. However, the extent to which results and ideas from group theory are used in these areas is often far less than the extent to which they could be used, sometimes resulting in a rediscovery of results in group theory from a different perspective. Similarly, many of the new directions that these subjects could potentially provide, are lost by group theory due to a lack of interaction.

One possible reason for this divide is the fact that in undergraduate coursework, the extent to which results in group theory are used to understand and provide perspective to other parts of mathematics is minimal. This can be contrasted with topology; we see the notions of compactness and connectedness, closed and open subsets, dense subsets, and other ideas from point-set topology, used routinely in analysis, algebraic geometry, Lie theory, differential geometry, and other parts of mathematics. From group theory, however, little is usually borrowed apart from the definition of group, subgroup, normal subgroup and the concept of group homomorphism.

The divide between group theory and mathematics has the following aspects:


 * People in other fields are unaware of certain facets of the language of group theory that could put results in those fields in better perspective. Described below are notions like permutably complemented subgroups, conjugacy-closed subgroups and conjugate-dense subgroup,s the close relation between Sylow theory and the theory of algebraic groups, and other examples.
 * People encountering groups in subjects that use mathematics may not have a very good idea of what group-theoretic or subgroup-theoretic conditions are strong and should be aimed for. Similar-looking and similar-sounding conditions in group theory could have very different levels of strength. For instance, the prime 2 dividing the order of a group is significantly different from the prime 3 dividing the order, as is witnessed in the odd-order theorem. Similarly, the existence of certain dihedral subgroups could have surprisingly strong implications on the structure of the group.

Facets of language in group theory
Group theorists have for a long time studied subgroup properties, some in the context of finite groups, some in the more general context of arbitrary groups. Some of these properties turn up in various guises as theorems in topology and linear algebra, but since the conceptual idea of the property is not known to people outside group theory, they may not be able to formalize the common ideas behind the various manifestations.

For example:


 * The notion of a characteristic subgroup, though well-known within group theory, is surprisingly unknown to people outside the subject, because preliminary courses in group theory often do not cover this notion. Yet, this notion is extremely useful in proving results about normal subgroups, and also has a lot of direct importance in geometric group theory.
 * Group theorists say that a subgroup has a normal complement if there is a normal subgroup intersecting it trivially and that generates with it the whole group. Famous results like the Schur-Zassenhaus theorem address the question of normal complements. Other people in group theory, working from a more geometric/topological/combinatorial perspective, use the term retract for a subgroup which possesses a normal complement. The latter tallies with the concept of retract in topology. Though these notions are clearly equivalent, they are developed in very separate ways.
 * Some group theory literature has considered the notion of a conjugacy-closed subgroup: a subgroup such that if two elements of the subgroup are conjugate in the group, they are conjugate in the subgroup. Though such subgroups have not been studied extensively in their own right, the related notion of fusion in a subgroup (conjugacy classes of the subgroup that merge in the whole group) is crucial to understanding finite groups. Interestingly, the concepts of conjugacy-closedness and fusion throw a lot of light on many results in linear algebra and topology. For a list of such examples, refer Category:Instances of conjugacy-closed subgroups.
 * There is a close relation between the Sylow theory in finite groups, and the theory of unipotent subgroups and Borel subgroups in algebraic groups and Lie groups.