examples of groups (original) (raw)

Groups (http://planetmath.org/Group) are ubiquitous throughout mathematics. Many “naturally occurring” groups are either groups of numbers (typically Abelian) or groups of symmetries (typically non-Abelian).

Groups of numbers

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  The integers modulo n, often denoted by ℤn, form a group under addition. Like ℤ itself, this is a cyclic group; any cyclic group is isomorphic to one of these.
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  The rational (or real, or complex) numbers form a group under addition.
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  The positive rationals form a group under multiplication with identity element 1, and so do the non-zero rationals. The same is true for the reals and real algebraic numbers.
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  More generally, any (skew) field gives rise to two groups: the additive group of all field elements with 0 as identity element, and the multiplicative group of all non-zero field elements with 1 as identity element.
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  The complex numbers of absolute value 1 form a group under multiplication, best thought of as the unit circle. The quaternions of absolute value 1 form a group under multiplication, best thought of as the three-dimensional unit sphere S3. The two-dimensional sphereS2 however is not a group in any natural way.
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  The positive integers less than n which arecoprime to n form a group if the operation is defined as multiplication modulo n. This is an Abelian group whose order is given by the Euler phi-function ϕ⁢(n).
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  The units of the number ring ℤ⁢[3] form the multiplicative group consisting of all integer powers of 2+3 and their negatives (see units of quadratic fields).
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  Generalizing the last two examples, if R is a ring with multiplicative identity 1, then the units of R (http://planetmath.org/GroupOfUnits) (the elements invertible with respect to multiplication) form a group with respect to ring multiplication and with identity element 1. See examples of rings.

Most groups of numbers carry natural topologies turning them into topological groups.

Symmetry groups

Other groups

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  The trivial group consists only of its identity element.
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  If X is a topological space and x is a point of X, we can define the fundamental group of X at x. It consists of (homotopy classes of) continuouspaths starting and ending at x and describes the structureof the “holes” in X accessible from x. The fundamental group is generalized by the higher homotopy groups.
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  Other groups studied in algebraic topology are the homology groups of a topological space. In a different way, they also provide information about the “holes” of the space.
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  The free groups are important in algebraic topology. In a sense, they are the most general groups, having only those relations among their elements that are absolutely required by the group axioms. The free group on the set S has as members all the finite strings that can be formed from elements of S and their inverses; the operation comes from string concatenation.
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  If A and B are two Abelian groups (or modules over the same ring), then the set Hom⁡(A,B) of all homomorphisms from A to B is an Abelian group. Note that the commutativity of B is crucial here: without it, one couldn’t prove that the sum of two homomorphisms is again a homomorphism.
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  Given any set X, the powerset 𝒫⁢(X) of X becomes an abelian group if we use the symmetric difference as operation. In this group, any element is its own inverse, which makes it into a vector space over ℤ2.
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  If R is a ring with multiplicative identity, then the set of all invertible n×n matrices over Rforms a group under matrix multiplication with the identity matrix as identity element; this group is denoted by GL⁡(n,R). It is the group of units of the ring of all n×n matrices over R. For a given n, the groups GL⁡(n,R) with commutative ring R can be viewed as the points on the general linear group scheme GLn.
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  The set of the equivalence classes of commensurability of the positive real numbers is an Abelian group with respect to the defined operation.
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  The set of arithmetic functions that take a value other than 0 at 1 form an Abelian group under Dirichlet convolution. They include as a subgroup the set of multiplicative functions.
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  Consider the curve C={(x,y)∈K2∣y2=x3-x}, where K is any field. Every straight line intersects this set in three points (counting a point twice if the line is tangent, and allowing for a point at infinity). If we require that those three points add up to zero for any straight line, then we have defined an abelian group structure on C. Groups like these are called abelian varieties.
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  Let E be an elliptic curve defined over any field F. Then the set of F-rational points in the curve E, denoted by E⁢(F), can be given the structure of abelian group. If F is a number field, then E⁢(F) is a finitely generated abelian group. The curve C in the example above is an elliptic curve defined over ℚ, thus C⁢(ℚ) is a finitely generated abelian group.
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  In the classification of all finite simple groups, several “sporadic” groups occur which don’t follow any discernable pattern. The largest of these is the monster group with about 8⋅1053elements.