We'd like to use analytics cookies to understand which symbols people look up, so we know what to research next. They are not used for advertising and we set none of them unless you say yes. What we store.
The notation of structure: groups, rings and fields, the arrows and corner marks of category theory, the invariants of topology, and the arithmetic functions of analytic number theory. What holds the group together is that its members encode a claim in a decoration rather than in a shape — a hook on an arrow's tail asserting injectivity, a corner inside a square asserting a universal property, a superscript rather than a subscript turning homology into cohomology. Several members exist mainly to warn: that G/N and L/K use one slash for two unrelated things, that the Legendre symbol is not a fraction, that (x)_n means opposite things in two literatures, that Γ(n) is off by one from n! and always has been.
A product sign with a short vertical bar on one side, marking which factor is normal.
A short vertical line with two side strokes, combining a group with a permutation action.
One group name over another separated by a slash, collapsing a subgroup to a point.
A group name inside vertical bars, counting its elements.
A capital Z or C with a subscript, naming the group generated by one element.
A capital S with a subscript, the group of all rearrangements of that many things.
A capital A with a subscript, the even permutations only.
A capital D with a subscript, the symmetries of a regular polygon.
The letters GL with a size and a field in brackets, the invertible matrices.
The letters SO with a dimension in brackets, the rotations of a space.
A capital U with a dimension in brackets, the complex analogue of rotation.
A generator inside angle or round brackets, naming everything it multiplies into.
Z over nZ, the integers wrapped round at a fixed size.
A ring name with a variable in square brackets, the polynomials over it.
One field over another separated by a slash, with no quotient implied.
Gal with a field extension in brackets, the symmetries of a polynomial's roots.
An arrow with a hooked tail, an injection.
An arrow with two heads, a surjection.
The letters id with an object subscript, the map that changes nothing.
A small right-angle bracket in a square diagram's corner, marking a fibre product.
A right-angle bracket in the opposite corner, marking an amalgamated sum.
A stylised h or よ, embedding a category into its functor category.
A repeated index, meaning a sum with no sigma written.
A lowercase epsilon with three indices, valued minus one, zero or one.
A lowercase delta with two indices, one when they match and zero otherwise.
A lowercase delta of a variable, infinitely tall and infinitely narrow.
A capital H or theta of a variable, zero then one.
A blackboard-bold 1 with a set subscript, one inside and zero outside.
Square brackets around two operators separated by a comma.
Curly braces around two operators separated by a comma.
A pair of curved curly braces around two functions in classical mechanics.
Square brackets around two vector fields or Lie algebra elements.
A stacked column of numbers in tall brackets, several rows deep.
A base with a bracketed or underlined superscript, a product of descending terms.
A base with a raised bracket or overline, a product of ascending terms.
An exclamation mark before a number instead of after it.
Two numbers stacked in tall brackets with no dividing line.
The Legendre symbol's glyph extended to composite denominators.
The letters lcm with a bracketed list, or square brackets alone.
A Greek phi of a number, counting coprime predecessors.
A Greek mu of a number, valued minus one, zero or one.
A Greek sigma with a subscript, summing powers of divisors.
A Greek pi of a number, counting primes rather than measuring circles.
A Greek zeta of a complex variable.
A capital Greek gamma of a variable, the factorial extended.
A capital Greek B of two variables, an integral of two powers.
A Greek psi of a variable, the gamma function's logarithmic derivative.
A Hebrew beth with a subscript, the tower of power sets.
Vertical bars around a set, counting its members.
A rounded double-cupped Greek omega, the first infinite ordinal.
A tilde over an equals sign, or a plain tilde, between two spaces.
A single wavy line, a weaker equivalence than homeomorphism.
A Greek pi with subscript one, a space and a base point.
A capital H with a degree subscript and a space in brackets.
A capital H with a raised degree, the dual of homology.
A curled ∂ applied to a chain or a region.
An overline on a set, adding its limit points.
A small circle raised after a set, or the word int.
A capital B with a centre and a radius in brackets.
A Greek chi of a space, a single whole number invariant.