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Sage_Note_Groups.xml
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<?xml version="1.0" encoding="UTF-8" ?>
<section xml:id="s-sage-groups-11">
<title>Using Sage to Study Groups</title>
<index><main>Sage Note</main><sub>Groups</sub></index>
<p>The group theory capabilities of Sage far exceeeds the scope of this book. In this section, we will concentrate on a few of the ideas we've discussed in this chapter.</p>
<p>Here is how to generate the group of integers modulo 23.</p>
<sage>
<input>
AdditiveAbelianGroup([23])
</input>
<output>
Additive abelian group isomorphic to Z/23
</output>
</sage>
<p>Here is a direct product with factors of the group of integers modulo 2.</p>
<sage>
<input>
G=AdditiveAbelianGroup([2,2,2])
G.list()
</input>
<output>
[(0, 0, 0), (1, 0, 0), (0, 1, 0), (1, 1, 0), (0, 0, 1), (1, 0, 1), (0, 1, 1), (1, 1, 1)]
</output>
</sage>
<p>Here is how to generate a cyclic group with 14 elements.</p>
<sage>
<input>
G=AbelianGroup(1,[14])
G.list()
</input>
<output>
(1, f, f^2, f^3, f^4, f^5, f^6, f^7, f^8, f^9, f^10, f^11, f^12, f^13)
</output>
</sage>
<p>There is no output from assigning <m>G</m>. The elements of <m>G</m> are generated from the <c>list</c> method. This group is isomorphic to \(\mathbb{Z}_{14}\). An isomorphism is suggested by the fact that when we multiply powers of \(f\), the exponents are added with \(+_{14}\). Among other things we can ask whether <m>G</m> is abelian and what
its subgroups are.</p>
<sage>
<input>
G.is_abelian()
</input>
<output>
True
</output>
</sage>
<p>The subgroups are identified by the groups to which they are isomorphic.</p>
<sage>
<input>
G.subgroups()
</input>
<output>
[Multiplicative Abelian subgroup isomorphic to C2 x C7 generated by {f}, Multiplicative Abelian subgroup isomorphic to C7 generated by {f^2}, Multiplicative Abelian subgroup isomorphic to C2 generated by {f^7}, Trivial Abelian subgroup]
</output>
</sage>
<p>We focus on the second of these subgroups. Recall that the index for the second subgroup is 1 since all lists start with the 0 index in Sage.</p>
</section>