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

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## Main Question or Discussion Point

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

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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_QED and Math Amateur
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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

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