# Group

# Group

A group is a simple mathematical system, so basic that groups appear wherever one looks in mathematics. Despite the primitive nature of a group, mathematicians have developed a rich theory about them. Specifically, a group is a mathematical system consisting of a set G and a binary operation *, which has the following properties:

[1] x*y is in G whenever x and y are in G (closure).

[2] (x*y)*z = x*(y*z) for all x, y, and z in G (associative property).

[3] There exists an element e in G such that e*x = x*e = x for all x in G (existence of an identity element).

[4] For any element x in G, there exists an element y such that x*y = y*x = e (existence of inverses).

Note that commutativity is not required. That is, it need not be true that x*y = y*x for all x and y in G.

One example of a group is the set of integers, under the binary operation of addition. Here the sum of any two integers is certainly an integer, 0 is the identity, *-a* is the inverse of a, and addition is certainly an associative operation. Another example is the set of positive fractions, *m/n,* under multiplication. The product of any two positive fractions is again a positive fraction, the identity element is 1 (which is equal to 1/1), the inverse of *m/n* is *n/m,* and, again, multiplication is an associative operation.

The two examples we have just given are examples of commutative groups. These are also known as Abelian groups in honor of Niels Henrik Abel (1802-1829), a Norwegian mathematician who was one of the early users of group theory. For an example of a non-commutative group consider the permutations on the three letters a, b, and c. All six of them can be described by

I is the identity; it sends *a* into *a, b* into *b,* and *c* into *c*. P then sends *a* into *a, b* into *c,* and *c* into *b.* Q sends a into *b, b* into *a,* and *c* into *c* and so on. Then P* Q = R since P sends *a* into *a* and Q then sends that *a* into *b*. Likewise P sends *b* into *c* and Q then sends that *c* into *c*. Finally, P sends *c* into *b* and Q then sends that *b* into *a*. That is the effect of first applying P and then Q is the same as R.

Following the same procedure, we find that Q* = S, which demonstrates that this group is not commutative. A complete “multiplication” table is:

From the fact that I appears just once in each row and column we see that each element has an inverse. Associativity is less obvious but can be checked. (Actually, the very nature of permutations allows us to check associativity more easily.) Among each group there are subgroups—subsets of the group which themselves form a group. Thus, for example, the set consisting of I and P is a subgroup since P* P = I. Similarly, I and T form a subgroup.

Another important concept of group theory is that of isomorphism. For example, the set of permutations on three letters is isomorphic to the set of symmetries of an equilateral triangle. The concept of isomorphism occurs in many places in mathematics and is extremely useful in that it enables us to show that some seemingly different systems are basically the same.

The term “group” was first introduced by the French mathematician Evariste Galois in 1830. His work was inspired by a proof by Abel that the general equation of the fifth degree is not solvable by radicals.

## Resources

### BOOKS

Bell, E.T. *Men of Mathematics*. Simon and Schuster, 1961.

Grossman, Israel, and Wilhelm Magnus. *Groups and Their Graphs.* Mathematical Association of America, 1965).

### OTHER

*University of St. Andrews, Scotland, School of Mathematics and Statistics: The Mac Tutor History of Mathematics Archives*. “The Development of Group Theory” <http://www-history.mcs.st-andrews.ac.uk/HistTopics/Development_group_theory.html> (accessed November 26, 2006).

Roy Dubisch

# Group

# Group

A group is a simple mathematical system, so basic that groups appear wherever one looks in **mathematics** . Despite the primitive nature of a group, mathematicians have developed a rich theory about them. Specifically, a group is a mathematical system consisting of a set G and a binary operation * which has the following properties:

*x*y*is in G whenever*x*and*y*are in G (closure).*(x*y)*z*=*x*(y*z)*for all*x, y,*and*z*in G (**associative property**).- There exists and element,
*e*, in G such that*e*x=x*e=x*for all*x*in G (existence of an**identity element**). - For any element
*x*in G, there exists an element*y*such that*x*y=y*x=e*(existence of inverses).

Note that commutativity is not required. That is, it need not be true that *x*y=y*x* for all *x* and *y* in G.

One example of a group is the set of **integers** , under the binary operation of **addition** . Here the sum of any two integers is certainly an integer, 0 is the identity, *-a* is the inverse of *a*, and addition is certainly an associative operation. Another example is the set of positive fractions, *m/n*, under **multiplication** . The product of any two positive fractions is again a positive fraction, the identity element is 1 (which is equal to 1/1), the inverse of *m/n* is *n/m*, and, again, multiplication is an associative operation.

The two examples we have just given are examples of commutative groups. (Also known as Abelian groups in honor of Niels Henrik Abel, a Norwegian mathematician who was one of the early users of group theory.) For an example of a non-commutative group consider the permutations on the three letters *a, b,* and *c*. All six of them can be described by

I is the identity; it sends *a* into *a*, *b* into *b*, and *c* into *c*. P then sends *a* into *a*, *b* into *c*, and *c* into *b*. Q sends *a* into *b*, *b* into *a*, and *c* into *c* and so on. Then P***Q=R since P sends *a* into *a* and Q then sends that *a* into *b*. Likewise P sends *b* into *c* and Q then sends that *c* into *c*. Finally, P sends *c* into *b* and Q then sends that *b* into *a*. That is the effect of first applying P and then Q is the same as R.

Following the same procedure, we find that Q***P=S which demonstrates that this group is not commutative. A complete "multiplication" table is as follows:

I | P | Q | R | S | T | |

I | I | P | Q | R | S | T |

P | P | I | R | Q | T | S |

Q | Q | S | I | T | P | R |

R | R | T | P | S | I | Q |

S | S | Q | T | I | R | P |

T | T | R | S | P | Q | I |

From the fact that I appears just once in each row and column we see that each element has an inverse. Associativity is less obvious but can be checked. (Actually, the very nature of permutations allows us to check associativity more easily.) Among each group there are *subgroups*-subsets of the group which themselves form a group. Thus, for example, the set consisting of I and P is a subgroup since P***P=I. Similarly, I and T form a subgroup.

Another important concept of group theory is that of *isomorphism*. For example, the set of permutations on three letters is isomorphic to the set of *symmetries* of an equilateral triangle. The concept of isomorphism occurs in many places in mathematics and is extremely useful in that it enables us to show that some seemingly different systems are basically the same.

The term "group" was first introduced by the French mathematician Evariste Galois in 1830. His work was inspired by a proof by Abel that the general equation of the fifth degree is not solvable by radicals.

## Resources

### books

Bell, E.T. *Men of Mathematics.* Simon and Schuster, 1961.

Grossman, Israel, and Wilhelm Magnus. *Groups and Their Graphs.* Mathematical Association of America, 1965.

Roy Dubisch

# group

group / groōp/ •
n. [treated as sing. or pl.] a number of people or things that are located close together or are considered or classed together: *these bodies fall into four distinct groups.* ∎ a number of people who work together or share certain beliefs: *I now belong to my local drama group.* ∎ a commercial organization consisting of several companies under common ownership. ∎ a number of musicians who play popular music together. ∎ Mil. a unit of the U.S. Air Force, consisting of two or more squadrons. ∎ Mil. a unit of the U.S. Army, consisting of two or more battalions. ∎ Art two or more figures or objects forming a design. ∎ Chem. a set of elements occupying a column in the periodic table and having broadly similar properties arising from their similar electronic structure. ∎ Chem. a combination of atoms having a recognizable identity in a number of compounds. ∎ Math. a set of elements, together with an associative binary operation, that contains an inverse for each element and an identity element. ∎ Geol. a stratigraphic division consisting of two or more formations.•
v. [tr.] (often be grouped) put together or place in a group or groups: *three wooden chairs were grouped around a dining table.* ∎ put into categories; classify: *we group them into species merely as a convenience.* ∎ [intr.] form a group or groups:

*many growers began to group together to form cooperatives.*

# group

**group** A set *G* on which there is defined a dyadic operation ◦ (mapping *G* × *G* into *G*) that satisfies the following properties:

(a) ◦ is associative;

(b) ◦ has an identity, i.e. there is a unique element *e* in *G* with the property that *x *◦ *e* = *e *◦ *x* = *x*

for all *x* in *G*; *e* is called the *identity* of the group;

(c) *inverses* exist in *G*, i.e. for each *x* in *G* there is an inverse, denoted by *x*^{–1}, with the property that *x *◦ *x*^{–1} = *x*^{–1} ◦ *x* = *e*

These are the *group axioms*.

Certain kinds of groups are of particular interest. If the dyadic operation ◦ is commutative, the group is said to be a *commutative group* or an *abelian group* (named for the Norwegian mathematician Niels Abel).

If there is only a finite number of elements *n* in the group, the group is said to be *finite*; *n* is then the *order* of the group. Finite groups can be represented or depicted by means of a Cayley table.

If the group has a generator then it is said to be *cyclic*; a cyclic group must be abelian.

The group is a very important algebraic structure that underlies many other algebraic structures such as rings and fields. There are direct applications of groups in the study of symmetry, in the study of transformations and in particular permutations, and also in error detecting and error correcting as well as in the design of fast adders.

Groups were originally introduced for solving an algebraic problem. By group theory it can be shown that algorithmic methods of a particular kind cannot exist for finding the roots of a general polynomial of degree greater than four. See also semigroup.

# group

**group, social group** A number of individuals, defined by formal or informal criteria of membership, who share a feeling of unity or are bound together in relatively stable patterns of interaction. The latter criterion is necessary in order to distinguish social groups from other aggregates dealt with by sociologists which are grouped only in the statistical sense that they share some socially relevant characteristic (including, for example, social categories such as suburban residents or junior managers). However, the term is one of the most widely used in sociology, and will often be found applied to combinations of people who may or may not share a feeling of unity (as in social class groups) and may or may not be involved in regular social interaction (as in the case of members of certain ethnic groups). See also COOLEY, CHARLES HORTON; DESCENT GROUPS; DYAD; GROUP DYNAMICS; OUT-GROUP; PARIAH GROUP; PEER GROUP; PRESSURE GROUPS; REFERENCE GROUP; STATUS GROUP; SUMNER, WILLIAM GRAHAM; TRIAD.

# group

# Group

# Group

a set of things collected as a unit. See also **gathering.**

*Examples:* group of columns [three or four columns joined together on the same pedestal], 1731; of company, 1748; of crystals, 1830; of islands; of musicians; of partisans, 1809; of rocks, 1859; of singers; of trees; of woes, 1729; of words, 1748.

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