I know there are infinitely many proper subgroups of $\mathbb{Z} \oplus \mathbb{Z}$, but is $\mathbb{Z} \oplus \mathbb{Z}$ cyclic?
The full problem is as stated in the title. I am here to check if this is a valid proof. I thought it would be easiest using Linear Algebra.     Recall that an infinite cyclic group is isomorphic to $\mathbb{Z}$. We wish to show that we do not have an isomorphism between $\mathbb{ZxZ\;and\;Z}$. Note that $\mathbb{ZxZ}$ is an infinite group (under addition of course). Now, in order for there to even be potential for an isomorphism, two spaces must have equal dimension. Since the $\mathbb{dim(\mathbb{ZxZ})}=2&gt;\mathbb{dim(\mathbb{Z})}=1$, we know that $\nexists$ an isomorphism between our spaces. Hence, $\mathbb{ZxZ}$ is not a cyclic group.   My question (besides a validity check): was there a better way to prove this? I just found this to be the easiest way.  EDIT Totally wrong with this one. Back to the cutting board.
Toolbar options are not displaying in Edit Profile - About me section How can I install Ubuntu 13.04 + without USB, CD, DVD or Net? (hard drive only) I have an older i386 Sony Vaio desktop PC where the BIOS only allows boot from floppy, CD or very specific USB device types, none of which seem to recognize USB flash drives for some reason. Floppy is not an option because the drive no longer works and I have no media for it if it did. I've tried assorted CD and DVD drives, the CD drives all give read errors and DVD drives all say "unsupported emulation".  My next possible option that I would like to try, is to install the hard drive into another machine as a second HD and figure out a way to format it and save an Ubuntu installation to run from that HD once it is moved back into the target machine. IS this possible and if so, how would I go about doing so? The currently running machine uses Windows XP. ActiveResource timeout not functioning How to resolve merge conflicts in Git repository? Why are .NET timers limited to 15 ms resolution? Is it possible to install Ubuntu through network? I would like to install Ubuntu  for almost 120 users. But it's very difficult in using USB and drives.  Any suggestions? Why energy curves spacetime? This  explains how energy curves spacetime. But I would like to know why energy curves spacetime, is there something like particle interaction that produce this result? Vertex of a pentagon-Does the algorithm always stop? To each vertex of a pentagon,we assign an integer $x_{i}$ with sum $s=\sum x_{i}&gt;0$. If x,y,z are the numbers assigned to 3 successive vertices and if $y&lt;0$,then we replace $(x,y,z)$ by $(x+y,-y,y+z)$.This step is repeated as long as there is a y&lt;0.Decide if the algorithm always stops. Generate uniformly distributed weights that sum to unity? It is common to use weights in applications like mixture modeling and to linearly combine basis functions. Weights $w_i$ must often obey $w_i ≥$ 0 and $\sum_{i} w_i=1$. I'd like to randomly choose a weight vector $\mathbf{w} = (w_1, w_2, …)$ from a uniform distribution of such vectors.  It may be tempting to use $w_i = \frac{\omega_i}{\sum_{j} \omega_j}$ where $\omega_i \sim$ U(0, 1), however as discussed in the comments below, the distribution of $\mathbf{w}$ is not uniform.  However, given the constraint $\sum_{i} w_i=1$, it seems that the underlying dimensionality of the problem is $n-1$, and that it should be possible to choose a $\mathbf{w}$ by choosing $n-1$ parameters according to some distribution and then computing the corresponding $\mathbf{w}$ from those parameters (because once $n-1$ of the weights are specified, the remaining weight is fully determined).  The problem appears to be similar to the  (but, rather than picking 3-vectors whose $ℓ_2$ norm is unity, I want to pick $n$-vectors whose $ℓ_1$ norm is unity).  Thanks! Could it be that Goldbach conjecture is undecidable? The result closest to Goldbach conjecture is Chen's theorem [Sci. Sinica 16 157–176], the proposition ``1+2''. It is natural to ask if it is likely that under our arithmetic axioms the Goldbach conjecture is an undecidable proposition. One problem I have found when learning calculus is that there are many different ways to denote the derivative. If $y=f(x)=x^2$, then we could write \begin{align} f'(x)&amp;=2x \\ y'&amp;=2x \\ \frac{df}{dx}(x)&amp;=2x \\ \frac{df(x)}{dx}&amp;=2x \\ \frac{d}{dx}f(x)&amp;=2x \\ \frac{dy}{dx}&amp;=2x \end{align} And this is just Lagrange and Leibniz's notations alone. What I find troubling is that they all seem to be suggesting subtly different things about what the derivative actually is. Is it a function, a limit of a quotient, or both? In the interests of keeping my post brief, I'll focus my attention on $f'(x)=2x$ and $\frac{dy}{dx}=2x$, as these seem to be the most common notations.  $$ f'(x)=2x $$  It does make sense to think of the derivative as the gradient function: $$ f'\colon x\mapsto\lim_{\Delta x \to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x} $$ In this case the limit expression is equal to $2x$, and so we can write $$ f' \colon x \mapsto 2x $$ However, this notation seems a little counter-intuitive when we consider what it means to differentiate a function with respect to a variable other than $x$. If I ask what is the derivative of $f(x)$ with respect to $\frac{x}{2}$, does this question make sense? Is it simply $f'(\frac{x}{2})$? Or do we have to express $x^2$ in terms of $\frac{x}{2}$? And how can we can express this derivative using Lagrange's notation?  $$ \frac{dy}{dx}=2x $$  There are many things which are nice about Leibniz's notation, including the fact that it is explicit which variable you are differentiating with respect to. However, in this case, it is unclear whether we are talking about a function, or something else entirely. There are other issues. Some people say they dislike the Leibniz formulation of the chain rule $$ \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx} $$ saying that they find it to be inaccurate. I don't really understand why this is the case. Could someone please elaborate? Geometric interpretation of $\frac {\partial^2} {\partial x \partial y} f(x,y)$ Are there any geometric interpretation for the second partial derivative? i.e.  $$f_{xy} = \frac {\partial^2 f} {\partial x \partial y}$$  In particular, I'm trying to understand the determinant from  for determining whether a critical point is a minima/maxima/saddle points:  $$D(a, b) = f_{xx}(a,b) f_{yy}(a,b) - f_{xy}(a,b)^2$$  I have no trouble understanding $f_{xx}(x,y)$ and $f_{yy}(x,y)$ as the of measure of concavity/convexity of f in the direction of x and y axis. But what does $f_{xy}(x,y)$ means? How to change the title of ToC? I need to change default title of ToC. Is it possible with titlesec package? Here is my MWE:  \documentclass{book}  \usepackage{titlesec} \titleformat{\chapter}{\normalfont\Large\scshape}{\thechapter}{1em}{} \vspace{6pt}]  \usepackage{lipsum}   %-------------------------------------------------------------------------  \begin{document}   \newpage \thispagestyle{empty} \tableofcontents \thispagestyle{empty} \newpage \thispagestyle{empty}  \setcounter{footnote} {0}\chapter*{{\fontsize{11}{14}\selectfont\normalfont{Author1} \\ {\scshape\Large Title}}} \addcontentsline{toc}{chapter}{\normalsize\normalfont{Author1} \\ \normalfont\scshape{Title}} \thispagestyle{empty} \Author{Author1} \Title{Title} \lipsum \newpage \thispagestyle{empty}    \end{document} How do popular free RAW editor/converter compare to each other on Windows? I'm looking for a free RAW editor/converter on Windows. Can you tell me some strong/weak points of them comparing with Capture NX and/or Adobe Photoshop Elements?       - conversion &amp; EXIF  - white balance  - registration not possible anymore   EXIF editing would be a nice bonus.  Related: How do I transfer programs from my old computer to my new Windows 7 computer? I know about the Easy Transfer tool included in Windows 7 that will transfer all my files but I want to transfer my programs. Also, I don't want to buy any gimmicky program if I don't have to. What early-2000s show about dimension hopping has a character who can swap limbs? An earlier question asked about a show that made me think of another one I can't remember.  It was a live action show about exploring different dimensions, and I think the main character was searching for his dad.  I remember seeing it in the US, but I don't know if it was limited to that.  I'm a little fuzzy on the details, but I think the main character could swap his limbs or something like that. Why is there no plural 's' after "hundred" or "thousand" in cardinal numbers? Let's say we have the numeral "12,345". Why is it pronounced as "twelve thousand three hundred and forty five" as opposed to "twelve thousands three hundreds and forty five". Are 's's omitted while pronouncing numerals to make the whole thing more succinct and easier to pronounce? Forward/Inverse Search Does Not work on certain tex files I have this weird issue with forward and inverse searches. I am using WinEdt + Sumatra combination. The forward and inverse search works perfectly on certain some .tex files, but fails on some others.  Currently, I am working on a .tex file and the inverse search always points to the end of a section. Where ever I double click in a particular section of the pdf, the inverse search points to the end of that section in the .tex file.  Moreover, when I try to perform forward search (Shift+F8) from the .tex file, the section in the pdf file doesn't get highlighted.  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