What is the difference between field and group in algebra

Question

What is the difference between a field and a group in algebra?

Could someone briefly explain it?

Answer 1

This is more of a rejoinder to PEV's response as it existed before the edit. One can think of "subtraction" as an operation separate from addition, though rather than consider the binary operation, one usually considers the unary operation that maps an element to its additive inverse (it has nicer properties than subtraction, which is not associative, and it automatically captures the idea that the additive inverse is unique).

In fact, from the point of view of Universal Algebra, we want a group to have three operations: a binary operation $\cdot$ (that corresponds to the group product), a unary operation ${}^{-1}$ (that maps every element to its inverse), and a zeroary operation $e$ (distinguished element). Then the "axioms" of a group can be stated as identities of the operations:

1. $(a\cdot b)\cdot c = a\cdot(b\cdot c)$;
2. $e\cdot a = a\cdot e = a$;
3. $a\cdot a^{-1} = a^{-1}\cdot a = e$.

Being expressible as identities has all sorts of nice consequences. Consequently, universal algebraists will consider a group as a set together with three operations.

Fields are more problematical, though. You start the same way: with two binary operations $+$ and $\times$; a unary operation $-$; two zeroary operations $0$ and $1$, and the axioms that describe these operations. Multiplicative inverses, however, are not defined via an operation, because $0$ does not have a multiplicative inverse. Instead, they must be defined via a partial operations, so fields are not "algebras" within the sense of universal algebra, but rather they are partial algebras. We know that fields cannot be defined as algebras with operations, because if they could then the category of fields (or even of fields of a given characteristic) with field morphisms would necessarily have categorical products, and they do not.

The theory of partial algebras is far more difficult than the theory of algebras. I seem to recall Brian Davey and David Clark, in their book Natural Dualities for the Working Mathematician mentioning that one can prove a lot of very nice and powerful "theorems" about partial algebras if one forgets some of the problems (e.g., if $f\colon A\to B$ is a homomorphism of partial algebras, then $f(A)$ need not be a partial-subalgebra of $B$; though this is particularly problematical when one consider relational algebras and relational partial algebras, where we have relations in addition to operations).

Answer 2

I feel this question is too broad for MSE, and could easily be answered by using a common search engine. Briefly: A field has two operations and is always commutative. See Definiton of Group and Definition of Field.

Answer 3

Another attempt to answer this question in more basic terms:

Groups and Fields are both important subjects of contemporary algebra. They are now recognized as strongly related, and arising in many situations :

• fields contain two different groups as substructure that work together nicely
• so, in order to prove that something is a field, you can first try to prove it contains these two groups, then prove other further properties (such as distributivity).

But originally, they came up from different questions.

• Groups arose from the study of substitutions (permutations) of symbols motivated by the theory of equations and the study of symmetry of geometric figures. They were then recognized as present in many aspects of mathematics.

• Fields arose as the generalization of the properties of the real numbers in regard to addition and multiplication, the usage of coordinates, and the expressive powers of formulas using symbols and basic operations.

But now mathematicians see them as both fundamentally linked together and to the ideas of number systems, sets of solutions of equations, solvability, the general concept of space, equations of physics. One particular kind of object which is very accessible to the beginner and can be an introduction to the links between the two are called: Finite Fields and one way to construct them is from the idea of the remainder of the division of an integer by a prime: in $29 = 7*4 + 1$, $1$ is called the remainder of the quotient of $29$ by $7$.

There is a broader concept relative to fields, called a ring, and there is the more global concept of algebra or algebraic structure which covers groups, rings and fields and many more. See also Varieties.

To have a broader idea of this family, you can lookup Lie algebras, Jordan algebras, Hopf algebras, von Neuman algebras, and further study a branch of algebra called Universal Algebra.

Answer 4

As Arturo points out, "a group has three operations with signature (2,1,0) (binary, unary, zeroary), while a field has two binary, one unary, two zeroary operations and one unary partial operation" (5 operations and 1 partial operation). A group has to satisfy (i) closure (ii) associativity, (iii) identity and (iv) invertibility. A field has to satisfy these as well and also have distributivity of multiplication over addition and commutativity. We can also talk about field extensions and group extensions. As Tao notes, one usually talks about field extensions as embedding a field $F$ into a larger field $K$ that has $F$ as a subspace. Group extensions, on the other hand, are defined by covering. That is, we cover a group $G$ by another group $G'$ that has $G$ has a quotient.