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'finitely generated group'
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| Title of object: |
finitely generated group |
| Canonical Name: |
FinitelyGenerated |
| Type: |
Definition |
| Created on: |
2002-02-03 01:36:48 |
| Modified on: |
2003-09-04 16:27:56 |
| Classification: |
msc:20A05 |
| Defines: |
finitely generated |
Revision comment (for changes between this and next version):
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
A group $G$ is {\em finitely generated} if there is a finite subset $X\subseteq G$ such that $X$ generates $G$. That is, every element of $G$ is a product of elements of $X$ and inverses of elements of $X$. Or, equivalently, no proper subgroup of $G$ contains $X$.
Every finite group is finitely generated, as we can take $X=G$.
Every finitely generated group is countable. |
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