# Search results

1. ### .Net Framework 3.5 on Windows 10 unable to load

Hi, A few days ago, my Windows 10 went through a long auto update and now when I run Windows Live mail, I cannot send an e-mail and I receive a message: An app on your PC needs the following Windows feature: .Net Framework 3.5 and then it instructs me to load it. But when I attempt to load...
2. ### Prove (Q/Z)/H is isomorphic to Q/Z

Homework Statement Let ##H## be a finite subgroup of the group ##G = \mathbb{Q}/\mathbb{Z}##. Prove ##G/H## is isomorphic to ##G##. Homework Equations [/B] My plan is to model the proof after a simple example but I would need to show there is a bijection between...
3. ### How to find composition series of Z_n^* of length k?

Hi, I'd like to find easily-accessible composition series of an interesting lengths say 10 or so (to study some theorems about subgroup series experimentally). I'm thinking the integer mod unit groups, that is ##\mathbb{Z}_n^*=Z_0\rhd Z_1\rhd Z_2\rhd \cdots \rhd Z_9\rhd Z_{10}=\{1\}##. Would...
4. ### How to find group types for a particular order?

Hi, How do we determine the group types for a particular order? I know the example of using the Cayley table to show there are only two types of groups of order 4 but do not know how to determine this for other groups. For example, suppose I wanted to show there is a group of order 10 in the...
5. ### Showing any transposition and p-cycle generate S_p

I was hoping someone could help me understand the following proof from: http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/genset.pdf The problem I'm having is the line Any p-cycle can be written as ##(1\;2\;\cdots\;p)## by relabeling the objects being permuted by applying an overall...
6. ### Anyone familiar with "GAP" for group arithemetic?

Hi guys, There is a software package called GAP for "Groups, Algorithms, and Programming" with emphasis on Group Theory. You can download it for free. I did. However I'm finding it so intractable to use. I would like to find the "missing group" in the Symmetric groups. That is, the group...
7. ### Converse of Lagrange's Theorem for groups

I know of only one group, ##A_4## of order 12 which does not have a subgroup with order dividing the group size. In this case, a subgroup of size ##6##. What property of a group causes this? Would I expect to find other examples only in non-abelian groups or are there abelian groups which do...
8. ### Do similar matrices respect multiplication

Homework Statement Let ##G=GL_n(F)## for ##F## a field, and define an equivalance relation by ##A\sim B## iff ##A## and ##B## are conjugate, that is, iff ##A=PBP^{-1}## for some ##P\in GL_n(F)##. Does ##\sim## respect multiplication? Homework Equations The equivalency respects...
9. ### Show group equivalence relation associated with normal subgroup

Homework Statement Let ##G## be a group and ##\sim## and equivalence relation on ##G##. Prove that if ##\sim## respects multiplication, then ##\sim## is the equivalence relation associated to some normal subgroup ##N\trianglelefteq G##; i.e., prove there is a normal subgroup ##N## such that...
10. ### How to find generators of symmetric groups

Hi, I was wondering how to find a minimal set of generators for the symmetric groups. Would it be difficult to fill-in the following table? ##\begin{array}{cl} S_3&=\big<(1\;2),(2\;3)\big> \\ S_4&=\big<(1\;2\;3\;4),(1\;2\;4\;3)\big>\\ \vdots\\ S_{500} \end{array} ## Is there a procedure to...
11. ### Examples of infinite nonabelian groups not GL_n(G)?

Hi, I was trying to identify some infinite non-abelian groups other than ##GL_n(G)## and also other than contrived groups such as the group: ##G=\big<r,s : r^2=s^3=1\big>## as per...
12. ### Normalizer N_G(X) equal to another set when o(X)<infty

1. The problem statement, all variables and given/known Prove that if a set ##X\subseteq G## is finite then it's Normalizer in G, ##N_G(X)## is also equal to the set: ##A=\{g\in G : gXg^{-1}\in X\}## Homework Equations Given ##X\subseteq G##, then the normalizer of X in G is defined as...
13. ### Software for drawing group lattice diagrams?

Hi, I was wondering if there is code already available to draw group lattice diagrams if I already know what the subgroup structure of the group and its subgroups are. For example, it's easy to determine the subgroup lattice for cyclic groups simply using divisors via Lagrange's Theorem...
14. ### Drawing 3D curved arrows around an axis

Hi, I'm working on a project to illustrate the isometries of a tetrahedron and would like to be able to add a curved arrow around the black axis of rotation in the Manipulate display to indicate which way the solid is rotating depending if the angle is negative or positive. Here's a display of...
15. ### Can quaternion group be represented by 3x3 matricies?

Hi, The Quaternion group, ##Q=\{1,-1,i,-i,j,-j,k,-k\}##, can be realized by ##2x2## matricies: ## \begin{align*} 1=\begin{bmatrix} 1,0 \\ 0,1\end{bmatrix} &\hspace{10pt} i=\begin{bmatrix} \omega,0 \\ 0,-\omega\end{bmatrix} & \hspace{10pt}j=\begin{bmatrix} 0,1 \\ -1,0\end{bmatrix} &...
16. ### Finding elements in GL_2(R) with specific orders

Hi, Is there a way to find elements in ##GL_2(\mathbb{R})## with an arbitrary order other than by trial and error? Suppose I wanted to find ##A\in GL_2(\mathbb{R}):\; o(A)=165##. Is there no methodical way to find this? Is it possible perhaps I can find a product of matricies, each...
17. ### Computing the order of n-folded automorphisms

I'd like to find ##\operatorname{aut}^2 \mathbb{Z}_n^*## and would first like to just compute it's order. However, I'm pretty sure in general ##\operatorname{aut}\mathbb{Z}_n^*## is non-abelian so can't apply or extend the formula for computing ##\big|\operatorname{aut}\mathbb{Z}_n^*\big|## to...
18. ### Why and how can one amino acid cause Bubonic plague?

What precisely is going on?
19. ### Mathematica Mathematica: Finding ideal generators as fast as possible

Hi, I'd like to find ideal'' generators for integer mod groups as quickly as possible and am running into timing problems when the number is large. It's a two-step process: (1) find orbits for all combinations of a generator set, and (2) select which generators are ideal. For (1), the step...
20. ### Mathematica Mathematica: supply list of list to Intersection?

I notice there are some functions in Mathematica in which you can't supply directly, an array of arguments but rather must give it a list. For example, GCD and LCM. I can use for example, GCD[2,3,8] and it returns 1. However if I code: GCD[{2,3,8}], the function call fails. I know of one way...
21. ### Mathematica Mathematica: create b-file for OEIS

I'd like to save a Mathematica array as a "b-file" so that I can upload it to the on-line encyclopedia of integer sequences. Here's a short version of my array: bData={{2,1},{3,1},{4,1},{5,2},{6,1},{7,2},{8,6},{9,2},{10,2},{11,4}} and the b-file format would be: 2,1 3,1 4,1 5,2 6,1 7,2 8,6...
22. ### Prove relation between generators and automorphisms of Z/nZ*

Hi, I was wondering if anyone here could help me prove or disprove this empirical observation or explain why there seems to be a connection between automorphic generators described below and the automorphisms of these groups: Consider the p-group expansion: ##\mathbb{Z}_n^*\cong S_2\times...
23. ### Do not wish courtesy "bump"

My thread was courtesy bumped and I feel that's immature, tacky, and unprofessional-looking. Now, I really hate complaining especially since I've had problems in the forum and would really rather just be a ghost, follow the rules and not bother anyone since PF is so beneficial to me and so many...
24. ### Finding non-trivial automorphisms of large Abelian groups

I feel there should be a way to find non-trivial (other than ##\operatorname{inn} G##) automorphisms of these groups other than by trial-and-error computation. Take for example ##\operatorname{aut}\mathbb{Z}_{100!}^*## . Are these just computationally inaccessable to us forever? How about a...
25. ### Find minimum number of generators for Z/nZ

Is there a method to find the minimum number of generators ##\big<a_1,a_2,\cdots,a_k\big>## needed to generate ##\mathbb{Z}_n^*## such that ##\mathop{\cap}\limits_{n=1}^k \big<a_n\big>=\{1\}## other than by just looking for them and checking orders and products of group elements? For example...
26. ### Mathematica Mathematica: unable to format isomorphic expression

I can't figure out how to format a variable isomorphic expression in Mathematica so that it will look nice. The best I can do will put parentheses around everything and expand powers. For example, given the array ##\{2,4,4,3,3\}##, I'd like Mathematica to write...
27. ### Finding number of automorphisms for Z/nZ^*

Hi, I'm studying Abstract Algebra and was wondering if someone could help me understand a formula for computing ##\big|\operatorname{aut} \mathbb{Z}_n^{*}\big|## using a formula described in this reference:http://www.msri.org/people/members/chillar/files/autabeliangrps.pdf Here is the basic...
28. ### You guys do pinnacles (yet)?

Well, I appear to be faced with one again. Like giving your PhD defense. That's a pinnacle I've not experienced. However, I've made dovetails which are another sort of pinnacle. I also rebuilt a 302. Another. I seem to want to do an electric service upgrade for my home and that is another...
29. ### Integral symbol for closed loops over functions?

I find it sometimes confusing dealing with integrals of multi-valued functions in distinguishing a closed integration path, and an integration path which forms a closed loop over the function. They can of course be quite different. For example: $$\oint f(z)dz.$$ Now, is the integration to...
30. ### Removing duplicate graphics in attachments

Hi, I was just wondering, in an effort to make the forum more classy-looking, and I know I already asked this question but it was never really answered by the administration, is there anyway to remove the redundant graphics attachments when I add an attachments, then go on to link that...