# 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:

• Some of the ali­ases and stuff around groups have been us­ing the term “alge­braic group”; this has a differ­ent tech­ni­cal mean­ing in math­e­mat­ics and should be avoided.