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


Well-known member
MHB Site Helper
Jun 22, 2012
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 [TEX] x \in X [/TEX] and [TEX] g \in G [/TEX], there is defined an element [TEX] g^x \in G [/TEX] such that if [TEX] g, h \in G [/TEX] then [TEX] {(gh)}^x = g^xh^x [/TEX]

I am not quite sure what the "operator set" is, but from what I can determine the notation [TEX] g^x [/TEX] 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.


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
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