Group theory

Group the­ory is the study of the alge­braic struc­tures known as “groups”. A group \(G\) is a col­lec­tion of el­e­ments \(X\) paired with an op­er­a­tor \(\bullet\) that com­bines el­e­ments of \(X\) while obey­ing cer­tain laws. Roughly speak­ing, \(\bullet\) treats el­e­ments of \(X\) as com­pos­able, in­vert­ible ac­tions.

Group the­ory has many ap­pli­ca­tions. His­tor­i­cally, groups first ap­peared in math­e­mat­ics as groups of “sub­sti­tu­tions” of math­e­mat­i­cal func­tions; for ex­am­ple, the group of in­te­gers \(\mathbb{Z}\) acts on the set of func­tions \(f : \mathbb{R} \to \mathbb{R}\) via the sub­sti­tu­tion \(n : f(x) \mapsto f(x - n)\), which cor­re­sponds to trans­lat­ing the graph of \(f\) \(n\) units to the right. The func­tions which are in­var­i­ant un­der this group ac­tion are pre­cisely the func­tions which are pe­ri­odic with pe­riod \(1\), and group the­ory can be used to ex­plain how this ob­ser­va­tion leads to the ex­pan­sion of such a func­tion as a Fourier se­ries \(f(x) = \sum \left( a_n \cos 2 \pi n x + b_n \sin 2 \pi n x \right)\).

Groups are used as a build­ing block in the for­mal­iza­tion of many other math­e­mat­i­cal struc­tures, in­clud­ing fields, vec­tor spaces, and in­te­gers. Group the­ory has var­i­ous ap­pli­ca­tions to physics. For a list of ex­am­ple groups, see the ex­am­ples page. For a list of the key the­o­rems in group the­ory, see the main the­o­rems page.

In­ter­pre­ta­tions, vi­su­al­iza­tions, and uses

Group the­ory ab­stracts away from the el­e­ments in the un­der­ly­ing set \(X\): the group ax­ioms do not care about what sort of thing is in \(X\); they care only about the way that \(\bullet\) re­lates them.\(X\) tells us how many el­e­ments \(G\) has; all other in­for­ma­tion re­sides in the choices that \(\bullet\) makes to com­bine two group el­e­ments into a third group el­e­ment. Thus, group the­ory can be seen as the study of pos­si­ble re­la­tion­ships be­tween ob­jects that obey the group laws; re­gard­less of what the ob­jects them­selves are. To vi­su­al­ize a group, we need only vi­su­al­ize the way that the el­e­ments re­late to each other via \(\bullet\). This is the ap­proach taken by group mul­ti­pli­ca­tion ta­bles and group di­a­grams.

Group the­ory is in­ter­est­ing in part be­cause the con­straints on \(\bullet\) are at a “sweet spot” be­tween “too lax” and “too re­stric­tive.” Group struc­ture crops up in many ar­eas of physics and math­e­mat­ics, but the group ax­ioms are still re­stric­tive enough to make groups fairly easy to work with and rea­son about. For ex­am­ple, if the or­der of \(G\) is prime then there is only one pos­si­ble group that \(G\) can be (up to iso­mor­phism). There are only 2, 2, 5, 2, and 2 groups of or­der 4, 6, 8, 9, and 10 (re­spec­tively). There are only 16 groups of or­der 100. If a group struc­ture can be found in an ob­ject, this makes the be­hav­ior of the ob­ject fairly easy to an­a­lyze (es­pe­cially if the or­der of the group is small). Group struc­ture is rel­a­tively com­mon in math and physics; for ex­am­ple, the solu­tions to a polyno­mial equa­tion are acted on by a group called the Galois group (a fact from which the un­solv­abil­ity of quin­tic polyno­mi­als by rad­i­cals was proven). Group the­ory is thus a use­ful tool for figur­ing out how var­i­ous math­e­mat­i­cal and phys­i­cal ob­jects be­have. For more on this idea, see the page on group ac­tions.

Roughly speak­ing, the con­straints on \(\bullet\) can be in­ter­preted as say­ing that \(\bullet\) has to treat the el­e­ments like they are a “com­plete set of trans­for­ma­tions” of a phys­i­cal ob­ject. The ax­iom of iden­tity then says “one of the el­e­ments has to be in­ter­preted as a trans­for­ma­tion that does noth­ing”; the ax­iom of in­ver­sion says “each trans­for­ma­tion must be in­vert­ible;” and so on. In this light, group the­ory can be seen as the study of pos­si­ble com­plete sets of trans­for­ma­tions. Be­cause the set of pos­si­ble groups is rel­a­tively limited and well-be­haved, group the­ory is a use­ful tool for study­ing the re­la­tion­ship be­tween trans­for­ma­tions of phys­i­cal ob­jects. For more dis­cus­sion of this in­ter­pre­ta­tion of group the­ory, see groups and trans­for­ma­tions.

Groups can also be seen as a tool for study­ing sym­me­try. Roughly speak­ing, given a set \(X\) and some re­dun­dant struc­ture atop \(X\), a sym­me­try of \(X\) is a func­tion from \(X\) to it­self that pre­serves the given struc­ture. For ex­am­ple, a physi­cist might build a model of two planets in­ter­act­ing in space, where an ar­bi­trary point picked out as the ori­gin. The be­hav­ior of the planets in re­la­tion to each other shouldn’t de­pend on which point they la­bel \((0, 0, 0)\), but many ques­tions (such as “where is the planet now?”) de­pend on the choice of ori­gin. To study facts that are true re­gard­less of where the ori­gin is (rather than be­ing ar­ti­facts of the rep­re­sen­ta­tion), they might let \(X\) be the pos­si­ble states of the model, with then ask what prop­er­ties of \(X\) are pre­served when the ori­gin is shifted — i.e., they ask “what facts about this model would still be true if I had picked a differ­ent ori­gin?” Trans­la­tion of the ori­gin is a “sym­me­try” of those prop­er­ties: It takes one state of \(X\) to an­other state of \(X\), and pre­serves all prop­er­ties of the phys­i­cal sys­tem that are in­de­pen­dent of the choice of ori­gin. When a model has mul­ti­ple sym­me­tries (such as ro­ta­tion in­var­i­ance as well as trans­la­tion in­var­i­ance), those sym­me­tries form a group un­der com­po­si­tion. Group the­ory tells us about how those sym­me­tries must re­late to one an­other. For more on this topic, see groups and sym­me­tries.

In fact, laws of physics can be in­ter­preted as “prop­er­ties of the uni­verse that are true at all places and all times, re­gard­less of how you la­bel things.” Thus, the laws of physics can be stud­ied by con­sid­er­ing the prop­er­ties of the uni­verse that are pre­served un­der cer­tain trans­for­ma­tions (such as chang­ing la­bels). Group the­ory puts con­straints on how these sym­me­tries in the laws of physics re­late to one an­other, and thus helps con­strain the search for pos­si­ble laws of physics. For more on this topic, see groups and physics.

Fi­nally, in a differ­ent vein, groups can be seen as a build­ing block for other math­e­mat­i­cal con­cepts. As an ex­am­ple, con­sider num­bers. “Num­ber” is a fairly fuzzy con­cept; the term in­cludes the in­te­gers \(\mathbb Z\), the ra­tio­nal num­bers \(\mathbb Q\) (which in­clude frac­tions), the real num­bers \(\mathbb R\) (which in­clude num­bers such as \(\sqrt{2}\)), and so on. The ar­row from \(\mathbb Z \to \mathbb Q \to \mathbb R\) points in the di­rec­tion of spe­cial­iza­tion: Each step adds con­straints (such as “now you must in­clude frac­tions”) and thus nar­rows the scope of study (there are fewer things in the world that act like real num­bers than there are that act like in­te­gers). Group the­ory is the re­sult of fol­low­ing the ar­row in the op­po­site di­rec­tion: groups are like in­te­gers but more per­mis­sive. Groups have an iden­tity el­e­ment that cor­re­sponds to \(0\) and an op­er­a­tion that cor­re­sponds to \(+\), but they don’t need to be in­finitely large and they don’t need to have any­thing cor­re­spond­ing to mul­ti­pli­ca­tion. From this point of view, “group” is a gen­er­al­iza­tion of “in­te­ger”, and from that gen­er­al­iza­tion, we can build up many differ­ent math­e­mat­i­cal ob­jects, in­clud­ing rings, fields, and vec­tor spaces. Em­piri­cally, many in­ter­est­ing and use­ful math­e­mat­i­cal ob­jects use groups as one of their build­ing blocks, and thus, some fa­mil­iar­ity with group the­ory is helpful when learn­ing about other math­e­mat­i­cal struc­tures. For more on this topic, see the tree of alge­braic struc­tures.

Children:

  • Group

    The alge­braic struc­ture that cap­tures sym­me­try, re­la­tion­ships be­tween trans­for­ma­tions, and part of what mul­ti­pli­ca­tion and ad­di­tion have in com­mon.

  • Group theory: Examples

    What does think­ing in terms of group the­ory ac­tu­ally look like? And what does it buy you?

  • Group action

    “Groups, as men, will be known by their ac­tions.”

Parents:

  • Mathematics

    Math­e­mat­ics is the study of num­bers and other ideal ob­jects that can be de­scribed by ax­ioms.