Exploring Abstract Algebra With Mathematica® by Allen C. Hibbard

By Allen C. Hibbard

• what's Exploring summary Algebra with Mathematica? Exploring summary Algebra with Mathematica is a studying surroundings for introductory summary algebra outfitted round a collection of Mathematica programs enti­ tled AbstractAlgebra. those programs are a beginning for this choice of twenty-seven interactive labs on workforce and ring conception. The lab section of this booklet displays the contents of the Mathematica-based digital notebooks con­ tained within the accompanying CD-ROM. scholars can have interaction with either the published and digital models of the fabric within the laboratory and search for information and reference details within the User's consultant. routines take place within the circulate of the textual content of labs, supplying a context during which to respond to. The notebooks are designed in order that the solutions to the questions can both be entered into the digital computing device or written on paper, whichever the trainer prefers. The notebooks help models 2. 2 and three. 0-4. zero and fit with all systems that run Mathematica. This paintings can be utilized to complement any introductory summary algebra textual content and isn't depending on any specific textual content. the crowd and ring labs were pass­ referenced opposed to a few of the extra well known texts. this knowledge are available on our site at http://www . significant. edu/eaarn. htrnl (which can be reflected at http://www . urnl. edu/Dept/Math/eaarn/eaarn. htrnl). in the event that your favourite textual content is not on our record, it may be extra upon request by way of contacting both author.

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1999 Subversively Grouping Our Elements 33 First let's consider a random group Zn for n in [6, 20]. To define this group, we need to first read in the Mathematica package that defines Zn and the other functions that we will be using. Needs["AbstractAlgebra'Master'I]; SwitchStructureTo[Group]; n G =Random [Integer, = Z [n] {6, 20}] Next we pick a random integer m, less than n, and then choose this many elements from G and put them in a set that we call H. ] ] }] H = RandomElements [G, m, Replacement -+ False] The question we would like to pursue first is whether this set H forms a subgroup of G, and if not, how we can make one with it.

Q12. Now suppose we have G = ZlO. What are the orders of the subgroups of G and how many subgroups are there of each order? Use the following cell if you want to do some experimenting. \n"}}] = = Q13. Now suppose we have G = Zll. What are the orders of the subgroups of G and how many are there of each order? Use the following cell if you want to do some experimenting. \n"}}] Q14. Summarize your findings by writing a conjecture about the subgroup structure of Zn. How might you prove your answer?

Q21. What is the order of Un? Give as complete an answer as possible, even if you don't have all the cases covered. 1 Prerequisites To complete this lab, you should be familiar with the definition of a subgroup of a group. 2 Goals What constitutes a subgroup? What elements are necessary before a set can be considered a subgroup? What do the subgroups of Zn look like? What about the subgroups of Un? What is the probability that a randomly chosen subset of elements from Zn will actually be a subgroup?

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