Sage: K = DihedralGroup ( 12 ) sage: sg = K. The factorization at the bottom might help you formulate a Sage will list each subgroup as a cyclic group with its generator. In the input box, enter the order ofĪ cyclic group (numbers between 1 and 40 are good initial choices) and Then this is a place where you can experiment with the structure of However, if you are viewing this as a worksheet in Sage, If you are viewing this as a PDF, you can safely skip over the nextīit of code. Special properties of cyclic groups (but see the next section). Usually list all of the subgroups of a group. conjugacy_classes_subgroups (), Subgroup of (Cyclic group of order 20 as a permutation group) generated by, Subgroup of (Cyclic group of order 20 as a permutation group) generated by, Subgroup of (Cyclic group of order 20 as a permutation group) generated by, Subgroup of (Cyclic group of order 20 as a permutation group) generated by, Subgroup of (Cyclic group of order 20 as a permutation group) generated by ]īe careful, this command uses some more advanced ideas and will not Sage: C20 = CyclicPermutationGroup ( 20 ) sage: C20. We can cut/paste an element of order 5 from the output above (in theĬase when the cyclic group has 20 elements) and quickly build a Sage: n = 20 sage: CN = CyclicPermutationGroup ( n ) sage: for g in CN. Group \(D_6\), since we will not keep repeating its definition below. Worksheet, be sure you have defined the group \(H\) as the dihedral OK, on to some popular command for groups. Helps to know some basic Python programming, but it is not required.) (To get the maximum advantage of using Sage it Riding on GAP’s IsAbelian() command and asking GAP do the You to determine that the is_abelian() function is basically For example, try H.is_abelian?, which will allow If you want to learn more about how Sage works, or possibly extend itsįunctionality, then you can start by examining the complete Python Is_abelian() function, describing the inputs and output, possibly This will display a portion of the source code for the H.is_abelian? (note the question mark) followed by the enter key. Find aįunction that looks interesting, say is_abelian(). Here is another couple of ways to experiment and explore. As before,Įxperiment and explore-it is really hard to break anything. You will get a list of available functions (you may need Sage: H = DihedralGroup ( 6 ) sage: H Dihedral group of order 12 as a permutation groupĪnd then a variety of functions become available.Īfter trying the examples below, experiment with tab-completion. Sage worksheet versions of this tutorial are available at Link below each cell, for a fully interactive experience. The worksheet version can be imported into the Sage notebookĮnvironment running in a web browser, and then the displayed chunks ofĬode may be executed by Sage if one clicks on the small “evaluate” This guide is also distributed in PDF format and as a Sage worksheet. The “E” in Sage stood for “Experimentation.” Not coincidentally, when Sage was the acronym SAGE, Order a student would learn the corresponding mathematics they mightīe encouraged to experiment and learn more about mathematics and learn Rather, by presenting commands roughly in the Texts on group theory and more information on Sage can be found via In an introductory course on group theory. This compilation collects Sage commands that are useful for a student 0 Version 1.3, dropped US on license, some edits.3 Version 1.1, added cyclic group size interact.
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