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Mathematical structures such as Numbers, Matrices, Vectors are all groups, rings, or similar?

Can you provide more examples of more thease.

Do operations such as addition occur (although in different ways) between any combination of two of thease, for example a number and a vector, or only between two of the same type, for example two vectors?

Is there a turm for the operations that can occur between groups? I think they are + and ×.

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Have you tried the Wikipedia article on group theory? – Qiaochu Yuan Jul 15 '12 at 15:44
Yes. I didn't have much luck with it. I'm looking for a simpler explanation, and I don't need to know how to use them mathematically, a high level description will do. – alan2here Jul 15 '12 at 17:05

Before answering your question, I think it will be helpful to make a point about what kind of thing a group is.

A group is a mathematical structure; it is in particular a collection of things $G$ (more formally known as a set), together with a binary operation $\star\;$, or in other words a rule that, to any elements $a,b$ of $G$, assigns some element $a\star b$ of $G$. This operation must satisfy some properties.

Thus, for example,

  • We can make a group out of the set of integers $\mathbb{Z}$ by choosing our operation to be $+\;$, i.e. $$a\star b\stackrel{\text{def}}{=}a+b.$$

  • We can make a group out of $\mathbb{R}^n$, the set of $n$-tuples of real numbers, by choosing our operation to be coordinate-wise addition, i.e. $$(a_1,\ldots,a_n)\star (b_1,\ldots,b_n)\stackrel{\text{def}}{=}(a_1+b_1,\ldots,a_n+b_n)$$

  • We can make a group out of $M_{m\times n}(\mathbb{R})$, the set of $m\times n$ matrices with real number entries, by choosing our operation to be entry-wise addition, i.e. $$\begin{bmatrix}a_{11} & \cdots & a_{1n}\\ \vdots & \ddots & \vdots\\ a_{m1} & \cdots & a_{mn}\end{bmatrix}\star\begin{bmatrix}b_{11} & \cdots & b_{1n}\\ \vdots & \ddots & \vdots\\ b_{m1} & \cdots & b_{mn}\end{bmatrix}\stackrel{\text{def}}{=}\begin{bmatrix}a_{11} +b_{11}& \cdots & a_{1n}+b_{1n}\\ \vdots & \ddots & \vdots\\ a_{m1}+b_{m1} & \cdots & a_{mn}+b_{mn}\end{bmatrix}$$

However, there are many groups where the operation is what would usually be referred to as multiplication, e.g.

  • We can make a group out of $\mathbb{R}\setminus\{0\}$, the set of non-zero real numbers, by choosing our operation to be multliplication, i.e. $a\star b\stackrel{\text{def}}{=}ab$.

and many groups where the collection does not consist of things that would ordinarily be thought of as numbers, e.g.

  • We can make a group out of the set of symmetries of a square by choosing our operation to be composition, i.e. doing one symmetry, and then another.

It is important to note that, when we speak about the collections of objects discussed above (and many others), it is implicitly assumed that we know that these particular choices of operations make them into a group. There are many other operations one could consider on the set of integers, or the set of matrices, etc., which do not make them into a group. It makes no sense to say that a set of objects $G$ is a group, because the operation on $G$ must be defined as well; but for common mathematical objects, the operation is not explicitly mentioned.

Thus, the elements of groups might be of any "type" you can think of. However, for each group $G$, its binary operation only has information regarding what to do with elements of $G$; without specifying what you mean, it would not make any more sense to try to add an integer and a vector than it would an integer and a symmetry of a square. However, we might very well decide to make a group of out of an integer, vector, and matrix; for example, if $$A=5\in\mathbb{Z},\qquad B=(1,2,3)\in\mathbb{R}^3,\qquad C=\begin{bmatrix} -3 & \pi \\ \sqrt{2} & 1\end{bmatrix}\in M_{2\times 2}(\mathbb{R})$$ we can define the collection $G=\{A,B,C\}$ and then define an operation $\star$ by $$\begin{array}{c|c|c|c|} \star & \mathbf A & \mathbf B & \mathbf C \\\hline \mathbf A & A & B & C\\\hline \mathbf B & B & C & A\\\hline \mathbf C & C & A & B\\\hline \end{array}$$ This operation satisfies all the necessary properties to make $G$ into a group. We have now defined a group where it does make sense to "add" (or, really, "star") things of different "types".

Lastly, and somewhat separately, I'd like to quote an earlier answer of mine regarding the term "number":

... there is no agreed-upon definition of what it means for something to be a "number". If someone said "I am only going to call even integers "numbers", the odd integers don't count", they are just as correct as someone who calls every mathematical construction humans have ever come up with a "number". Mathematicians have decided on definitions of "integer", "complex number", etc., and the statements that "0 is an integer" or "$\sqrt{2}$ is not an integer" are true because "integer" actually means something. The word "number" is a vague word that has no mathematical content.

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