Group

A group is an ab­strac­tion of a col­lec­tion of sym­me­tries of an ob­ject. The col­lec­tion of sym­me­tries of a tri­an­gle (ro­tat­ing by \(120^\circ\) or \(240^\circ\) de­grees and flip­ping), re­ar­range­ments of a col­lec­tion of ob­jects (per­mu­ta­tions), or ro­ta­tions of a sphere, are all ex­am­ples of groups. A group ab­stracts from these ex­am­ples by for­get­ting what the sym­me­tries are sym­me­tries of, and only con­sid­ers how sym­me­tries be­have.

A group \(G\) is a pair \((X, \bullet)\) where:

  • \(X\) is a set, called the “un­der­ly­ing set.” By abuse of no­ta­tion, \(X\) is usu­ally de­noted by the same sym­bol as the group \(G\), which we will do for the rest of the ar­ti­cle.

  • \(\bullet : G \times G \to G\) is a bi­nary op­er­a­tion. That is, a func­tion that takes two el­e­ments of a set and re­turns a third. We will ab­bre­vi­ate \(x \bullet y\) by \(xy\) when not am­bigu­ous. This op­er­a­tion is sub­ject to the fol­low­ing ax­ioms:

  • Clo­sure: \(\bullet\) is a func­tion. For all \(x, y\) in \(X\), \(x \bullet y\) is defined and in \(X\). We ab­bre­vi­ate \(x \bullet y\) as \(xy\).

  • Iden­tity: There is an el­e­ment \(e\) such that \(xe=ex=x\) for all \(x \in X\).

  • In­verses: For each \(x\) in \(X\), there is an el­e­ment \(x^{-1} \in X\) such that \(xx^{-1}=x^{-1}x=e\).

  • As­so­ci­a­tivity: \(x(yz) = (xy)z\) for all \(x, y, z \in X\).

1) The set X is the col­lec­tion of ab­stract sym­me­tries that this group rep­re­sents. “Ab­stract,” be­cause these el­e­ments aren’t nec­es­sar­ily sym­me­tries of some­thing, but al­most all ex­am­ples will be.

2) The op­er­a­tion \(\bullet\) is the ab­stract com­po­si­tion op­er­a­tion.

3) The ax­iom of clo­sure is re­dun­dant, since \(\bullet\) is defined as a func­tion \(G \times G \to G\), but it is use­ful to em­pha­size this, as some­times one can for­get to check that a given sub­sets of sym­me­tries of an ob­ject is closed un­der com­po­si­tion.

4) The ax­iom of iden­tity says that there is an el­e­ment \(e\) in \(G\) that is a do-noth­ing sym­me­try: If you ap­ply \(\bullet\) to \(e\) and \(x\), then \(\bullet\) sim­ply re­turns \(x\). The iden­tity is unique: Given two el­e­ments \(e\) and \(z\) that satisfy ax­iom 2, we have \(ze = ez = z.\) Thus, we can speak of “the iden­tity” \(e\) of \(G\). This jus­tifies the use of \(e\) in the ax­iom of in­ver­sion: ax­ioms 1 through 3 en­sure that \(e\) ex­ists and is unique, so we can refer­ence it in ax­iom 4.

\(e\) is of­ten writ­ten \(1\) or \(1_G\), be­cause \(\bullet\) is of­ten treated as an ana­log of mul­ti­pli­ca­tion on the set \(X\), and \(1\) is the mul­ti­plica­tive iden­tity. (Some­times, e.g. in the case of rings, \(\bullet\) is treated as an ana­log of ad­di­tion, in which case the iden­tity is of­ten writ­ten \(0\) or \(0_G\).)

5) The ax­iom of in­verses says that for ev­ery el­e­ment \(x\) in \(X\), there is some other el­e­ment \(y\) that \(\bullet\) treats like the op­po­site of \(x\), in the sense that \(xy = e\) and vice versa. The in­verse of \(x\) is usu­ally writ­ten \(x^{-1}\), or some­times \((-x)\) in cases where \(\bullet\) is analo­gous to ad­di­tion.

6) The ax­iom of as­so­ci­a­tivity says that \bul­let be­haves like com­po­si­tion of func­tions. When com­pos­ing a bunch of func­tions, it doesn’t mat­ter what or­der the in­di­vi­d­ual com­po­si­tions are com­puted in. When com­pos­ing \(f\), \(g\), and \(h\), we can com­pute \(g \circ f\), and then com­pute \(h \circ (g \circ f)\), or we can com­pute \(h \circ g\) and then com­pute \((h \circ g) \circ f\), and we will get the same re­sult.

knows-req­ui­site(Monoid): Equiv­a­lently, a group is a monoid which satis­fies “ev­ery el­e­ment has an in­verse”.

knows-req­ui­site(Cat­e­gory the­ory): Equiv­a­lently, a group is a cat­e­gory with ex­actly one ob­ject, which satis­fies “ev­ery ar­row has an in­verse”; the ar­rows are viewed as el­e­ments of the group. This jus­tifies the in­tu­ition that groups are col­lec­tions of sym­me­tries. The ob­ject of this cat­e­gory can be thought of an ab­stract ob­ject that the iso­mor­phisms are sym­me­tries of. A func­tor from this cat­e­gory into the cat­e­gory of sets as­so­ci­ates this ob­ject with a set, and each of the mor­phisms a per­mu­ta­tion of that set.

Examples

The most fa­mil­iar ex­am­ple of a group is per­haps \((\mathbb{Z}, +)\), the in­te­gers un­der ad­di­tion. To see that it satis­fies the group ax­ioms, note that:

  1. (a) \(\mathbb{Z}\) is a set, and (b) \(+\) is a func­tion of type \(\mathbb Z \times \mathbb Z \to \mathbb Z\)

  2. \((x+y)+z=x+(y+z)\)

  3. \(0+x = x = x + 0\)

  4. Every el­e­ment \(x\) has an in­verse \(-x\), be­cause \(x + (-x) = 0\).

For more ex­am­ples, see the ex­am­ples page.

Notation

Given a group \(G = (X, \bullet)\), we say ”\(X\) forms a group un­der \(\bullet\).” \(X\) is called the un­der­ly­ing set of \(G\), and \(\bullet\) is called the group op­er­a­tion.

\(x \bullet y\) is usu­ally ab­bre­vi­ated \(xy\).

\(G\) is gen­er­ally al­lowed to sub­sti­tute for \(X\) when dis­cussing the group. For ex­am­ple, we say that the el­e­ments \(x, y \in X\) are “in \(G\),” and some­times write “\(x, y \in G\)” or talk about the “el­e­ments of \(G\).”

The or­der of a group, writ­ten \(|G|\), is the size \(|X|\) of its un­der­ly­ing set: If \(X\) has nine el­e­ments, then \(|G|=9\) and we say that \(G\) has or­der nine.

Resources

Groups are a ubiquitous and use­ful alge­braic struc­ture. When­ever it makes sense to talk about sym­me­tries of a math­e­mat­i­cal ob­ject, or phys­i­cal sys­tem, groups pop up. For a dis­cus­sion of group the­ory and its var­i­ous ap­pli­ca­tions, re­fer to the group the­ory page.

A group is a monoid with in­verses, and an as­so­ci­a­tive loop. For more on how groups re­late to other alge­braic struc­tures, re­fer to the tree of alge­braic struc­tures.

Children:

  • Order of a group
  • Abelian group

    A group where the op­er­a­tion com­mutes. Named af­ter Niels Hen­rik Abel.

  • Group: Examples

    Why would any­one have in­vented groups, any­way? What were the his­tor­i­cally mo­ti­vat­ing ex­am­ples, and what ex­am­ples are im­por­tant to­day?

  • Group: Exercises

    Test your un­der­stand­ing of the defi­ni­tion of a group with these ex­er­cises.

  • Group homomorphism

    A group ho­mo­mor­phism is a “func­tion be­tween groups” that “re­spects the group struc­ture”.

  • Cyclic group

    Cyclic groups form one of the most sim­ple classes of groups.

  • Symmetric group

    The sym­met­ric groups form the fun­da­men­tal link be­tween group the­ory and the no­tion of sym­me­try.

  • Group isomorphism

    “Iso­mor­phism” is the proper no­tion of “same­ness” or “equal­ity” among groups.

  • Order of a group element
  • Dihedral group

    The dihe­dral groups are nat­u­ral ex­am­ples of groups, aris­ing from the sym­me­tries of reg­u­lar poly­gons.

  • Group conjugate

    Con­ju­ga­tion lets us perform per­mu­ta­tions “from the point of view of” an­other per­mu­ta­tion.

  • Normal subgroup

    Nor­mal sub­groups are sub­groups which are in some sense “the same from all points of view”.

  • Alternating group

    The al­ter­nat­ing group is the only nor­mal sub­group of the sym­met­ric group (on five or more gen­er­a­tors).

  • Group coset
  • Simple group

    The sim­ple groups form the “build­ing blocks” of group the­ory, analo­gously to the prime num­bers in num­ber the­ory.

  • Prime order groups are cyclic

    This is the first step on the road to clas­sify­ing the finite groups.

  • Lagrange theorem on subgroup size

    La­grange’s The­o­rem is an im­por­tant re­stric­tion on the sizes of sub­groups of a finite group.

  • Cauchy's theorem on subgroup existence

    Cauchy’s the­o­rem is a use­ful con­di­tion for the ex­is­tence of cyclic sub­groups of finite groups.

  • Group orbit
  • Cyclic Group Intro (Math 0)

    A finite cyclic group is a lit­tle bit like a clock.

  • Subgroup

    A group that lives in­side a big­ger group.

  • Group presentation

    Pre­sen­ta­tions are a fairly com­pact way of ex­press­ing groups.

  • Every group is a quotient of a free group
  • Free group

    The free group is “the purest way to make a group con­tain­ing a given set”.

Parents: