It is often convenient to write this as an infinite product over all the primes, where all but a finite number have a zero exponent.

It is known that the least order of a finite group for which there exists two commutators whose product is not a commutator is 96 ; in fact there are two nonisomorphic groups of order 96 with this property.

* Compact Lie groups are all known: they are finite central quotients of a product of copies of the circle group S < sup > 1 </ sup > and simple compact Lie groups ( which correspond to connected Dynkin diagrams ).

Each term consists of the product of a constant ( called the coefficient of the term ) and a finite number of variables ( usually represented by letters ), also called indeterminates, raised to whole number powers.

If G is a finite group and S and T are subgroups of G, then ST is a subset of G of size | ST | given by the product formula:

The open sets in the product topology are unions ( finite or infinite ) of sets of the form, where each U < sub > i </ sub > is open in X < sub > i </ sub > and U < sub > i </ sub > ≠ X < sub > i </ sub > only finitely many times.

However, despite also being called the direct sum of rings when I is finite, the product of rings is not a coproduct in the sense of category theory.

Given a Hilbert space L < sup > 2 </ sup >( m ), m being a finite measure, the inner product < ·, · > gives rise to a positive functional φ by

** Atomic domain, an integral domain in which every non-zero non-unit is a finite product of irreducible elements

One may define the Cartesian product of any finite collection of sets recursively:

Formally, we start with a category C with finite products ( i. e. C has a terminal object 1 and any two objects of C have a product ).

Any finite product in a preadditive category must also be a coproduct, and conversely.

In category theory and its applications to mathematics, a biproduct of a finite collection of objects in a category with zero object is both a product and a coproduct.

In a preadditive category the notions of product and coproduct coincide for finite collections of objects.

If C is a preadditive category, then every finite product is a biproduct, and every finite coproduct is a biproduct.

Notice that each coefficient in the product AB only depends on a finite number of coefficients of A and B.

If I is an index set and X < sub > I </ sub > is the set of indeterminates X < sub > i </ sub > for i ∈ I, then a monomial X < sup > α </ sup > is any finite product of elements of X < sub > I </ sub > ( repetitions allowed ); a formal power series in X < sub > I </ sub > with coefficients in a ring R is determined by any mapping from the set of monomials X < sup > α </ sup > to a corresponding coefficient c < sub > α </ sub >, and is denoted.

The only angles of finite order that may be constructed starting with two points are those whose order is either a power of two, or a product of a power of two and a set of distinct Fermat primes.

A free group on a set S is a group where each element can be uniquely described as a finite length product of the form:

If we then let N be the subgroup of F generated by all conjugates x < sup > − 1 </ sup > R x of R, then it is straightforward to show that every element of N is a finite product x < sub > 1 </ sub >< sup > − 1 </ sup > r < sub > 1 </ sub > x < sub > 1 </ sub >.

More generally, if S is a subset of a group G, then < S >, the subgroup generated by S, is the smallest subgroup of G containing every element of S, meaning the intersection over all subgroups containing the elements of S ; equivalently, < S > is the subgroup of all elements of G that can be expressed as the finite product of elements in S and their inverses.

( 2003, 2005 ) put methods such as the Strassen and Coppersmith – Winograd algorithms in an entirely different group-theoretic context, by utilising triples of subsets of finite groups which satisfy a disjointness property called the triple product property ( TPP ).

Since the square root of an algebraic integer is again an algebraic integer, it is not possible to factor any nonzero nonunit algebraic integer into a finite product of irreducible elements, which implies that is not Noetherian!

Given two fractional ideals I and J, one defines their product IJ as the set of all finite sums: the product IJ is again a fractional ideal.

But even if that other plant employs the same number of workers and makes the same product, there are other facts to consider.

Alpha decay is by far the most common form of cluster decay where the parent atom ejects a defined daughter collection of nucleons, leaving another defined product behind ( in nuclear fission, a number of different pairs of daughters of approximately equal size are formed ).

A number of formal and industry standards exist for bicycle components to help make spare parts exchangeable and to maintain a minimum product safety.

The inner product of two vectors is a complex number.

# for each line, the number of product groups is equal to.

Now setting all of the X < sub > s </ sub > equal to the unlabeled variable X, so that the product becomes, the term for each k-combination from S becomes X < sup > k </ sup >, so that the coefficient of that power in the result equals the number of such k-combinations.

While at university, Gauss independently rediscovered several important theorems ; his breakthrough occurred in 1796 when he showed that any regular polygon with a number of sides which is a Fermat prime ( and, consequently, those polygons with any number of sides which is the product of distinct Fermat primes and a power of 2 ) can be constructed by compass and straightedge.

The type of end product resulting from a condensation polymerization is dependent on the number of functional end groups of the monomer which can react.

Furthermore, if b < sub > 1 </ sub > and b < sub > 2 </ sub > are both coprime with a, then so is their product b < sub > 1 </ sub > b < sub > 2 </ sub > ( modulo a it is a product of invertible elements, and therefore invertible ); this also follows from the first point by Euclid's lemma, which states that if a prime number p divides a product bc, then p divides at least one of the factors b, c.

The total amount of scattering in a sparse medium is determined by the product of the scattering cross-section and the number of particles present.

This space is homeomorphic to the product of a countable number of copies of the discrete space S.

For example, a sales transaction can be broken up into facts such as the number of products ordered and the price paid for the products, and into dimensions such as order date, customer name, product number, order ship-to and bill-to locations, and salesperson responsible for receiving the order.

Here n is the number of electrons / mole product, F is the Faraday constant ( coulombs / mole ), and ΔE is cell potential.

In number theory, the fundamental theorem of arithmetic, also called the unique factorization theorem or the unique-prime-factorization theorem, states that every integer greater than 1 is either prime itself or is the product of prime numbers, and that, although the order of the primes in the second case is arbitrary, the primes themselves are not.

In algebraic number theory 2 is called irreducible ( only divisible by itself or a unit ) but not prime ( if it divides a product it must divide one of the factors ).

The gamma function must alternate sign between the poles because the product in the forward recurrence contains an odd number of negative factors if the number of poles between and is odd, and an even number if the number of poles is even.

The number 54 can be expressed as a product of two other integers in several different ways: