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 \(\displaystyle \supseteq \). 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

Note 1: This has also been posted on MHF

Note 2: I tried to place a pdf of pages 142-146 inclusive on MHB but since it was 220kb it would not upload. Readers interested can see the pdf on MHF

 

[jvicens]

Dear Peter,

Thank you for your inquiry about Isaacs' treatment of X-Groups in his book Algebra - A Graduate Course. To answer your first question, Isaacs' approach to Noetherian modules is not the most common approach, but it is a valid and accepted way to study module theory. His terminology, however, is not standard and may not be widely used in other books on the subject.

In general, a Noetherian module is defined as a module that satisfies the ascending chain condition (ACC) on submodules. This means that there are no infinite increasing chains of submodules. Isaacs' definition of a Noetherian X-group is a variation of this, where the module is considered with respect to a specific set X and the poset of all X-groups. This is not a common approach, but it is a valid one.

As for the terminology "X-Group," it is not a standard term in module theory. It appears that Isaacs has coined this term to refer to a group with a specific set X and the operations defined on it. In other books, such a group may be referred to as a "module with an action" or a "module with operators." The notation g^x may also vary in other sources, but it typically represents the action of x on g.

To give you an example of an X-group, let's consider the group G = {1, 2, 3, 4, 5} with the set X = {a, b, c}. We can define the following operations on G:

1^a = 2, 2^a = 3, 3^a = 4, 4^a = 5, 5^a = 1
1^b = 3, 2^b = 4, 3^b = 5, 4^b = 1, 5^b = 2
1^c = 4, 2^c = 5, 3^c = 1, 4^c = 2, 5^c = 3

In this example, G is an X-group with X = {a, b, c}. We can see that the operation of x on g is simply shifting the elements in G to the right, with the last element wrapping around to the first. This is just one example, and there may be many different