I've seen groups, rings, and fields described with a multiplication operation as well as a group defined as only having addition and subtraction (via inverse) operations. Is the reason the answer varies with respect to a group having or not having a multiplication operation dependent upon the type of numbers represented (e.g. integers, reals, etc) as well as the elements included (e.g. 0 is not in Z_p* because it doesn't have an inverse? Or do groups never have a multiplication operation?
Is multiplication as an operation available in groups, rings and fields over Z_p*
abstract-algebrafinite-groupsgroup-theory
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I think this is a legitimate observation -- it is possible to do lots of group theory without knowing much about integers or real numbers, but these number systems seem to come up immediately whenever one is dealing with rings.
It seems to me that the main reason is that every ring has a characteristic, and therefore a ring with unity necessarily includes a copy of either the integers or some $\mathbb{Z}_n$. Furthermore, any element of a finitely-generated ring can be written as a (possibly noncommutative) polynomial in the generators, where the coefficients of the polynomial are integers (or perhaps some elements of $\mathbb{Z}_n$). Adding and multiplying these polynomials inherently requires addition and multiplication of integers.
Once you get past this basic "number-ness", there are many rings that are otherwise unrelated to either integers or real numbers. For example, it's possible to develop the theory of polynomial rings quite a bit without using numbers for anything other than coefficients. The same goes for group rings, whose elements are hardly more number-like than typical group elements. For a more sophisticated example, the elements of the (integer) cohomology ring of a topological space are essentially geometric objects, not numbers.
Fields are the same way. Every field inherently contains either the rational numbers $\mathbb{Q}$ or a prime field $\mathbb{Z}_p$, but aside from that one could argue that many fields have very little to do with numbers.
Yes: if $F$ is a field, then "module over $F$" is the same thing as "vector space over $F$". Said another way: vector spaces are modules in which the ring of scalars is a field.
A "linear algebra" (or more generally, an "$F$-algebra") is both a ring and an $F$-vector space, in such a way that the ring multiplication is compatible with the $F$-vector space structure.
You have "group actions" if all you have is a set and a map $G\times X\to X$ which is compatible with the operations of $G$. If $X$ has an algebraic structure of its own, e.g., if $X$ is an abelian group, then we talk about $G$-modules (which amounts to having a group homomorphism $G\to\mathrm{Aut}(X)$, where "Aut" are the appropriate structure automorphisms).
All of these can be further generalized to the concept of "general/universal algebra" (fields are not universal algebras, but they can be obtained by weakening the conditions to obtain 'partial algebras'). A great introduction to that is George Bergman's An Invitation to General Algebra and Universal Constructions.
Best Answer
By definition of group there is only one binary operation required to have certain properties (associativity, existence of an identity element, inverses).
However there is a convention to write the group operation as addition (+) if the operation is commutative (we say the group is Abelian), and more generally when the group is not commutative (or we don't know) to write the group operation as multiplication.
This is only a convention. The group axioms for the binary operation will work with any symbol for it, so if it helps to think of it as multiplication, you are not wrong. In one important family of examples the group elements are symmetries or (stated another way) mappings that preserve a set of things (e.g. permutations), and in those cases the "multiplication" is actually composition of functions, symbolized by $\circ$.
There are many algebraic structures which have a group operation connected to them. Vector spaces, for example, have a commutative group operation called vector addition. A division ring $\langle D,+,* \rangle$ has a commutative addition and a (possibly) noncommutative multiplication such that the nonzero elements have inverses and thus form a (possibly) noncommutative group (so the case of a division ring with commutative multiplication is a field).
But we should also note that many times we use "multiplication" to mean a binary operation that does not have all the nice properties of a group operation. If we drop the requirement of an identity and inverses, and keep only the associative property and "closure" (that the result of the binary operation is defined), that sort of algebraic structure is called a semigroup. The multiplication of an arbitrary ring forms a semigroup. By dropping associativity (leaving only the closure property), one defines a magma.
These may seem awfully abstract ideas, but "strange" binary operations often arise from the study of more familiar ones. For instance, when the entries of a square $n\times n$ matrix are taken from a ring, we can define matrix multiplication. But even when the ring is a field, the matrix multiplication so defined will in general only give us a semigroup (or, if the ring is assumed to have a multiplicative unit, the matrices form a monoid, i.e. a semigroup with an identity element). So one is led to these generalizations (and specializations) by natural applications.