Abelian X-Groups and Noetherian (Abelian) X-Groups

[Math Amateur]

I was having a quick look at Isaacs : Algebra - A Graduate Course and was interested in his approach to Noetherian modules. I wonder though how standard is his treatment and his terminology. Is this an accepted way to study module theory and is his term X-Group fairly standard (glimpsing at other books it does not seem to be!) and, further, if the structure he is talking about is a standard item of study, is his terminology "X-Group" standard? If not, what is the usual terminology.

A bit of information on Isaacs treatment of X-Groups follows:

In Chapter 10: Operator Groups and Unique Decompositions, on page 129 (see attachment) Isaacs defines an X-Group as follows:

0.1 DEFINITION. Let X be an arbitrary (possibly empty) set and Let G be a group. We say that G is an X-group (or group with operator set X) provided that for each $x \in X$ and $g \in G$, there is defined an element $g^x \in G$ such that if $g, h \in G$ then ${(gh)}^x = g^xh^x$

I am not quite sure what the "operator set" is, but from what I can determine the notation $g^x$ refers to the conjugate of g with respect to x (this is defined on page 20 - see attachment)

In Chapter 10: Module Theory without Rings, Isaacs defines abelian X-groups and uses them to develop module theory and in particular Noetherian and Artinian X-groups.

Regarding a Noetherian (abelian) X-group, the definition (Isaacs page 146) is as follows:

DEFINITION. Let M be an abelian X-group and consider the poset of all X-groups ordered by the inclusion $\supseteq$. We say M is Noetherian if this poset satisfies the ACC (ascending chain condition)

My question is - is this a standard and accepted way to introduce module theory and the theory of Noetherian and Artinian modules and rings.

Further, can someone give a couple of simple and explicit examples of X-groups in which the sets X and G are spelled out and some example operations are shown.

Peter

 

[R136a1]

It's not really a standard way to do module theory, but it certainly is a standard way of doing the Jordan-Holder theorem. If we would not do it that way, then we would need different theorems for different occasions.

Specific examples:

If $G$ is any group and $X=\emptyset$, then an X-group is just the same as a group.

If $G$ is any group and if $X=\{2\}$ (or another number), then we get groups where $(gh)^2 = g^2 h^2$. These are abelian groups.

If $G$ is any group and $X=G$, then we can set $g^x = xgx^{-1}$, the conjugation.

If $M$ is a vector space over $\mathbb{R}$, then take $X=\mathbb{R}$. The usual scalar multiplication then gives $g^x$.

The notion of X-groups is used most often in settings about dimensions, or series of subgroups.

 

[Math Amateur]

Do you know Martin Isaacs book in which he uses X-Groups for the Jordan-Holder Theorem and as a way to introduce modules.

How do you rate his book and approach (even through it is non-standard in terms of introducing module theory.

Peter

 

[R136a1]

I know Isaacs book, and I think it is an excellent book. So if you like it, then you should keep doing it.

 

[blue_raver22]

Thank you for your interest in Isaacs' approach to Noetherian modules. His treatment and terminology are not necessarily standard, but they are accepted and used by many mathematicians. The term "X-group" is not a commonly used term in module theory, but it is used by Isaacs to emphasize the role of operators in his approach. The usual terminology for this structure is "module with operators" or "module with a fixed endomorphism set" (see for example the book "Modules and Rings" by T.Y. Lam).

Isaacs' approach is a valid and interesting way to study module theory and Noetherian modules. However, it is not the only approach and there are other standard ways to define and study Noetherian and Artinian modules and rings. For example, in the book "Commutative Rings" by H. Matsumura, Noetherian and Artinian modules are defined using the descending chain condition and the ascending chain condition on submodules, respectively.

As for examples of X-groups, one simple example is the group of integers \mathbb{Z} with the set X being the set of prime numbers. In this case, the operator g^x is the conjugate of g by the prime number x. Another example is the group of 2x2 invertible matrices with real entries, denoted by GL(2,\mathbb{R}). The set X can be taken to be the set of all real numbers, and the operator g^x is the conjugate of g by the real number x. These examples may seem trivial, but they illustrate the concept of X-groups and how the operator set X can be used to define different structures on a group or module.

I hope this helps clarify Isaacs' approach and terminology. As with any mathematical topic, there are different ways to approach and study it, and Isaacs' approach is one valid and interesting way to do so.