Clarification on the order of a group I just need some clarification. I have a question stating that G is a group and x and y are elements of that group, both of which are order 2. So being that the order of G is finite, what would be the order of xy? It can be assumed that x and y are not inverses of each other. I think x^2 is the identity and y^2 is also the identity being as both of them are of order two. Am I thinking about this correctly?
Examples and further results about the order of the product of two elements in a group Let $G$ be a group and let $a,b$ be two elements of $G$. What can we say about the order of their product $ab$?   says "not much":     There is no general formula relating the order of a product $ab$ to the orders of $a$ and $b$. In fact, it is possible that both $a$ and $b$ have finite order while $ab$ has infinite order, or that both $a$ and $b$ have infinite order while $ab$ has finite order.   On the other hand no examples are provided. $(\mathbb{Z},+), 1$ and $-1$ give an example of elements of infinite order with product of finite order. I can't think of any example of the other kind! So:   What's an example of a group $G$ and two elements $a,b$ both of finite order such that their product has infinite order?     Wikipedia then states:     If $ab = ba$, we can at least say that $\mathrm{ord}(ab)$ divides $\mathrm{lcm}(\mathrm{ord}(a), \mathrm{ord}(b))$   which is easy to prove, but not very effective. So:   What are some similar results about the order of a product, perhaps with some additional hypotheses?
How can I port forward with iptables? I want connections coming in on ppp0 on port 8001 to be routed to 192.168.1.200 on eth0 on port 8080.  I've got these two rules  -A PREROUTING  -p tcp -m tcp --dport 8001 -j DNAT --to-destination 192.168.1.200:8080  -A FORWARD -m state -p tcp -d 192.168.1.200 --dport 8080 --state NEW,ESTABLISHED,RELATED -j ACCEPT   and it doesn't work. What am I missing? Unwanted faint lines connecting dashed double line Building on the answer  by wrtlprnft, I have the following code     \documentclass[tikz,margin=1cm]{standalone}    \usetikzlibrary{decorations.markings}    \begin{document}    \begin{tikzpicture}         \tikzstyle{arrowed double line}=[             dashed, double distance between line centers=3pt,             postaction=decorate,             decoration={                 markings,                 mark=between positions 10pt and -10pt step 20pt with {                     \arrow[thin,yshift= 1.5pt,xshift=.8pt]{&gt;}                     \arrow[thin,yshift=-1.5pt,xshift=.8pt]{&lt;}                 },             },         ]         \draw[arrowed double line] (0,0) --  (1,2);     \end{tikzpicture} \end{document}   which produces this:    I am unhappy with the faint transverse lines between the dashes. I don't even know why tikz decides to draw them.  Is there a way to get rid of them?  Thanks in advance to all replies.  PS the same thing occurs at the caps of double solid lines How can I get a more traditional drop-down like menu for Unity? How to count the number of set bits in a 32-bit integer? 8 bits representing the number 7 look like this:  00000111   Three bits are set.     What are algorithms to determine the number of set bits in a 32-bit integer? What does the term $\mathcal O(\epsilon^2)$ mean? Sum of two squares modulo p I have heard somewhere that for all primes $p$, for all $k$, there exist $x, y$ s.t. $x^2 + y^2\equiv k \pmod{p}$? I recall that the proof is very elementary, but I can't remember such a proof. How would one prove this? One way is to use Cauchy-Davenport, but I don't think that this is the simplest solution. I'm looking for a bijection from the closed interval [0,1] to the real line. I have already thought of $\tan(x-\frac{\pi}{2})$ and $-\cot\pi x$, but these functions aren't defined on 0 and 1.   Does anyone know how to find such a function and/or if it even exists?   Thanks in advance! How to reproduce this box in tcolorbox I'm working with tcolorbox, I saw the pdf  and I want to reproduce the little green box in this image:     Can anyone help me please? biblatex: displaying all authors of multi-author works in the bibliography I'm using the  package to add citations and a bibliography to my LaTeX document. I've noticed that only the first author plus "et al." is displayed for works with more than three authors. That's fine with me for in-text-citations, but I'd rather have the complete author information in the bibliography. How can I do that? Replace and Association If Z is a square integrable random variable such that $E[Z]&gt;0$, then prove that  $$P(Z\geq k) \geq \frac{(E(Z)-k)^2}{E(Z^2)} \forall k \in [0,E(Z)]. $$  This is a homework question and I don't even understand why the inequality is reversed comparing to Markov/Chebychev inequality. Should Cauchy-Schwartz inequality be used? If yes, then how to begin? Thanks a ton! Just curious as to how to obtain item drops from the last update.  To clarify, item drops as in weapon skins, crates, anything else Valve added without letting us know, etc.  It seems that after playing a bit, I've gotten no drops, but I've seen players who've gotten drops after a few short minutes of gameplay, but I've been playing for hours.  Is the item drop system like TF2, where you can just idle, get an item drop, come back? Is it like Dota2, where you have to be in the matchmaking system and complete matches for the chance to get drops? Or do you have to be MvP to get item drops? Is it a certain time you play the game?   Also, are weapon and crate drops independent or dependent on each other? Meaning, is the drop counter shared between Weapons and Crates, or are they both separate drop counters? Find all $x$ such that $x^6 = (x+1)^6$. Does a Magic Staff count as a magic weapon? Many creatures have resistance to weapons that aren't "magic weapons".  A Magic Staff can be used as a quarterstaff, but (usually) doesn't have a bonus to hit (a "plus").  Is it considered a Magic Weapon for the purpose of damage resistance? If $H\leq G$ and $[G:H]=2$, then $H$ is normal in $G$. An .mov video that is upside down in Windows appears right side up in OSX Reordering polygon shapefile permanently? I have a Transcend Storejet external USB hard disk. This is not an SSD, but a mechanical disk with rotating plates. I have formatted the whole disk, without partitioning, as NTFS. I have used mkntfs tool under Linux.  When plugged into a Linux machine, the system sees a drive with two partitions (/dev/sdc /dev/sdc1 /dev/sdc2). However, as I know it is a partitionless disk, I can mount the whole device (mount -t ntfs /dev/sdc /mnt) and it works without any problems.  When plugged into a MS Windows XP machine, the systems sees a disk with two unformatted partitions, and it does not assign a drive letter to either the whole disk, nor to any of the partitions.  Would anyone know how to get MS Windows to mount my disk as a NTFS superfloppy?  I have already tried removing old USB mounting points and residual devices using 'DriveCleanup'. It did not help.  By the way, I also have an external Kingston USB SSD, also formatted as NTFS superfloppy, without partitioning. However, this one is recognized and mounted normally by MS Windows. Are there any versions of t-SNE for streaming data? My understanding of  and the Barnes-Hut approximation is that all data points are required so that all force interactions can be calculated at the same time and each point can be adjusted in the 2d (or lower dimensional) map.  Are there any versions of t-sne that can efficiently deal with streaming data? So if my observations are arriving one at a time, it will find the best location on the 2d map to place the new observation, or continuously update all points on the 2d map to account for ht new observation.  Would this even make sense or does it go against the setup of t-sne. Why Changing Permissions For Some Systems Files is Not Allowed?