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

  1. #1
    Super Member Bernhard's Avatar
    Jan 2010
    Hobart, Tasmania, Australia

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

    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 $\displaystyle x \in X $ and $\displaystyle g \in G $, there is defined an element $\displaystyle g^x \ in G $ such that if $\displaystyle g, h \in G $ then $\displaystyle {(gh)}^x = g^xh^x $

    I am not quite sure what the "operator set" is, but from what I can determine the notation $\displaystyle 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.

    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.

    Last edited by Bernhard; Oct 24th 2013 at 08:33 PM.
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