I am working on proving that $z = r^{k}$ is the only nonidentity element of $D_{2n}$ ($n = 2k$) which commutes with all elements of $D_{2n}$. I have been able to show that any other power of $r$ cannot commute with $s$. To do this I showed that $$r^j s = sr^{n-j}$$ I understand that $r^j \neq r^{n-j}$, but how can it be assured that $r^j s \neq sr^{n-j}$? I have been reading a solution of this problem that claims from here it suffices to show that $sr^j$ does not commute with $r$ for any integer $j$. Why is this sufficient? How did they choose $sr^j$?
How to prove that the center of the dihedral group $D_{2n}$ is $\{1,r^{n}\}$ and the center of $D_{2n-1}$ is $\{1\}$?  I don't know how to prove it in this general case.
When you have a card like  which reads "Exile Target Creature with power 3 or less", as I understand it, it becomes untargetable if the opponent's creature power increases thanks to a pump spell like .  But what if the Disregard is already on the stack and targeting; and they respond with Brute Strength? I'm pretty sure (and my opponent with Brute Strength was certain) that it won't exile the creature, but I don't know which rule states this, and there's no gatherer ruling on the spell. Can you use your off-handed weapon for your main attack action? My friend is 2 weapon fighting a flesh golem. He has one sword that does lightning damage, and one sword that does fire damage. His right hand holds the lightning sword, and his left hand holds the fire sword.  My friend wants to attack with his left hand because lightning heals the golem. However, the DM says that because his left hand is his left hand, he can't use it as his "main" hand, and only attack with it. Is this rules as written/intended? How to Measure Luck vs Skill in Games? Find and replace text within multiple files I want to know how I can find and replace a specific text in multiple files like in Notepad++ in the linked tutorial.  e.g.: How do I unlock a SQLite database? sqlite&gt; DELETE FROM mails WHERE (`id` = 71); SQL error: database is locked   How do I unlock the database so this will work? Why didn't Dumbledore prevent the prosecution of Sirius Black? Dumbledore performed the Fidelius charm on the Potters, so he knew who the Secret-Keeper was1. That implies he knew not Sirius but Wormtail betrayed the Potters. Why didn't he prevent Sirius's conviction and suspect the explosion killing twelve Muggles was caused by Wormtail (and that Sirius cornered Wormtail instead of the other way around)?   1: Though in  it is said Dumbledore didn't know. I find it hard to believe the person casting the Fidelius charm doesn't know who the Secret-Keeper is. UNO Stacking a Draw 4 Card instead of picking up 4 cards? Need clarification on the Draw 4 rule. - Player 1 lays down a Draw 4 Card and says the color is Red. - Player 2 does not have a Red card but has a Draw 4 Card. Can Player 2 lay down his Draw 4 Card and have Player 3 draw 8 cards? $k[x]/(x^n)$ module with finite free resolution is free Where can I find out about Stack Overflow history? I'd like to know how Stack Overflow and other sites started - which was first, who initiated it (OK, that's on the link below) and why, how the different sites evolved, how the rules evolved, etc.  Where can I find such information?  Of course, I know I can look at  for a brief overview but I'm wondering if there is info on this site, particularly more detailed.. Why is the charge naming convention wrong? Prove that , any primitive root $r$ of $p^n$ is also a primitive root of $p$ For an odd prime $p$, prove that any primitive root $r$ of $p^n$ is also a primitive root of  $p$   So I have assumed $r$ have order $k$ modulo $p$ , So $k|p-1$.Then if I am able to show that $p-1|k$ then I am done .But I haven't been able to show that.Can anybody help me this method?Any other type of prove is also welcomed. Is there any way to create a bootable Ubuntu USB flash drive from the terminal without using any third-party applications like , , ,  etc.  I tried to create a bootable Ubuntu flash drive with dd method,  sudo umount /dev/sdb sudo dd if=/path/to/ubuntu.iso of=/dev/sdb bs=1M   It creates files on the USB disk, but when I try to boot the USB disk it shows an Operating System Not Found error. How do I update Android on my device? This Community Wiki question is designed to be the "canonical question" for questions asking how to upgrade one's device to a new Android version.   I want to get a newer version of the OS for my Android device. How can I do that?  See also: Same gravatar is displayed for different users in stackexchange.com filters Same gravatar is displayed for different users in the stackexchange.com , under the Top Network Askers in the right side bar.   The red circled image's source (&lt;img src=) are from   https://www.gravatar.com/avatar/?s=18&amp;d=identicon&amp;r=PG&amp;s=30                                ^   Seems the gravatar id is missing here in the image source.  For an example from the list, the user  gravatar id should be    https://www.gravatar.com/avatar/66c66c4d97a7bc3859a85253f003e1fb?s=18&amp;d=identicon&amp;r=PG&amp;s=30   but it is mapped with   https://www.gravatar.com/avatar/?s=18&amp;d=identicon&amp;r=PG&amp;s=30   which caused the display issue. Comparing negatives: "she seems not to know" vs. "she doesn't seem to know" What is the difference in style and meaning between the following two:        She seems not to know.   She doesn't seem to know.      Is there a name to this type of construction? How to understand what's taking up disk space? I'm looking for a linux alternative to . I would like to know what is taking up space on my hard drives.  A program that works on console and doesn't require a UI is preferred . Proving the identity $\sum_{k=1}^n {k^3} = \big(\sum_{k=1}^n k\big)^2$ without induction What is the safest way to clean up /boot partition? I have 200 MB assigned for the /boot partition.  Whenever I try to update the kernel, I receive an error message that basically states /boot is full.  What can I do to cleanup /boot and remove/backup the older kernels? Can I connect to a sandbox using JWT Flow and Prod connect app? I'm trying to connect to a sandbox using the sfdx cli and the JWT flow through force:auth:jwt:grant. According to , we can use the same connected app information from Production to connect to our Scratch Org.  Is it something that should work for sandbox too ? I'm getting a "Failed: Not approved" error.  Connected app in sandbox comes from the refresh but has a different consumer key. The Permission set is set correctly on my user.  Is there a way to connect to a recently refreshed sandbox using the JWT flow (without manually changing something like the consumer key) ? I would like to know if there is a word or phrase for a systematic campaign to annihilate every descendant of a particular ancestor.   We have the word 'genocide' for killing off an entire race, and plenty of words for killing people based on how they are related to you (fratricide, filicide, et. al.), but looking over the  and the , I couldn't find a term for systematically killing an entire bloodline.   My question is prompted by a US news story about the horrific murder of 8 members of the same family, but those murders are not an example of what I'm thinking of because the children of the victims were left alive, so the motive for the killings was not to kill off the bloodline. The sense that some group had coordinated an attack against just one family is what made me wonder what the "not an entire race, just one bloodline" version of 'genocide' would be called.   Searching the Internet for "kill off" and bloodline sends me straight into the land of vampires, Illuminati, reptilian overlords, and, naturally, Priory of Sion nonsense so I haven't had much luck finding a thread to pull.   It may be that there is no word for it in the sense of murdering a human bloodline, because we just don't attempt kill off our royals down to the last heir often enough to coin a term for it. Maybe there is a scientific term for culling a line from a test population as part of biological research?