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group – identity element
group
A set together with a binary operation defined on its elements, satisfying - The group operation is associative,
- There is an identity element, i.e., an element e of the group so that for any other element a of the group we have ea = ae = a, and
- for every element a of the group there is an element a´, called the inverse of a, so that aa´ = a´a = e.
If the group operation is commutative, the group is called Abelian. If the condition that every element have an inverse is dropped, the set is called a monoid. If in addition the requirement that an identity element exist is dropped, the set is called a semigroup. Finally, if the condition that the group operation be associative is dropped, the set is called a groupoid.
 Cf. field, module, ring
group automorphism
A group isomorphism from a group to itself. That is, a bijective function from a group to itself that preserves the group operation, i.e. f(ab) = f(a)f(b) for all a, b.
Cf. group homomorphism, group isomorphism.
group homomorphism
A function from one group to another that preserves the group operation, i.e., f(ab) = f(a)f(b) for all a, b.
Cf. group automorphism, group isomorphism.
group isomorphism
A group homomorphism that is both “one-to-one” and “onto.” That is, a bijective function from one group to another that preserves the group operation, i.e. f(ab) = f(a)f(b) for all a, b.
Cf. group homomorphism, group automorphism.
groupoid
See group.
harmonic series
The infinite series whose terms are the reciprocals of the natural numbers
This series diverges, i.e., the sum does not exist.
Related article: Series
Hausdorff space
A topological space is Hausdorff if every pair of distinct points have disjoint neighborhoods. Two disjoint compact sets in a Hausdorff space have disjoint neighborhoods. A compact subset of a Hausdorff space is closed.
Heawood Map Coloring Theorem
For every positive integer n, the chromatic number of the orientable surface of genus n (sphere with n handles) is given by
For n = 0, this statement is exactly the Four Color Theorem, which was proven later, and does not bear Heawood's name.
Heine-Borel property
Every closed and bounded subset of the real numbers is compact.
Hempel’s Ravens Paradox
ARTICLE
A paradox of inductive logic described by the philosopher C.G. Hempel. See the article for a complete exposition.
hexagon
A regular polygon having six equal sides and six equal interior angles. Hexagons can be used to tile the plane, and do so more efficiently than any other polygon.
Cf. tesselation.
Hilbert’s Problems
ARTICLE
23 problems posed by Hilbert to the community of mathematicians at the Paris conference of 1900. See the article for details.
homeomorphism
A homeomorphism is a bijective, continuous transformation of one topological space onto another whose inverse is also continuous.
homomorphism
A function f from one algebra to another is called a homomorphism if it preserves operations on elements, that is, if for any a, b in the domain, f(ab) = f(a)f(b).
Cf. group homomorphism, ring homomorphism.
hyperbola
The locus of points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.
Like the ellipse and parabola, the hyperbola is a conic section.
Related article: Conics
hyperset
A set which is not well-founded, i.e., which involves self-membership or, equivalently, an infinite descending membership chain. Example: the Quine atom x = {x}.
hypotenuse
On a right triangle, the side opposite the right angle.
i
See imaginary number.
icosahedron
A polyhedron having twenty faces.
The faces of a regular icosahedron are congruent, equilateral triangles.
Cf. Platonic solid.
idempotent
An element of an algebra is called idempotent if it is equal to its own square.
If for a binary operation “ · ” on an algebra A we have that a · a = a for every a in A, then “ · ” is called an idempotent operation.
identity element
Given an algebra X with a binary operation “ • ”, an identity element of X is an element e such that for all x in X, x • e = e • x = x. If “ • ” is addition, the identity element is usually called zero and denoted by 0; if the operation is multiplication, the identity element is usually called unity and denoted by 1.
It is possible for an algebra to have distinct left and right identities, that is, distinct e and e’ such that for all x, e • x = x and x • e’ = x. If an element is both a left and right identity, then it is the unique identity element in the algebra for that operation.
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