Problem: If $G$ has exactly one subgroup $H$ of order $k$, then $H$ is normal in $G$.  I can see the obvious low level solution, that $gHg^{-1}$ is also a subgroup of order $k$ and therefore $H$ is normal. What confuses me is that this problem was in the chapter titled "Homomorphisms". The author introduces the idea of a homomorphism, proves the isomorphism theorems, and then hands us this problem.   I was thinking to try to use the first isomorphism in an obvious way such that $H$ is the kernel of some homomorphism but I couldn't find it.   Can someone help me find the likely "intended" solution? Since the purpose of exercises should be to help me understand homomorphisms and ismorphisms better, my solution can't really be considered satisfactory.
How can I show that if some group $G$ has only one subgroup $K$ of order $n$, then $K$ is a normal subgroup? Would that mean that it only has one subgroup total? If so then I guess that makes sense.
The integral $\int_0^8 \sqrt{x^4+4x^2}\,dx$ What the reason the number of lines differs?  $ head -n 100000 ./access.log &gt; ./data/log.sample $ cat $_ | wc -l 1933424 Deleting polygon defined area from raster layer in ArcGIS Desktop? Which tool do I use in ArcGIS 10.1 for Desktop to exclude areas covered by vector polygon layers from a raster layer?   I want to keep portions of the raster layers that exclude the portions covered by the vector layer. What race (species?) is Star Lord's dad? At the end of the new Guardians of the Galaxy Vol. 2 Trailer we meet Star Lord's father:                                  Dad appears to be human. But Peter Quill's mother said in the first film that his father was a      "being made of pure light".    What race (species?) is Star Lord's father in the MCU? Splitting raster into smaller chunks using GDAL? I have a raster (USGS DEM actually) and I need to split it up into smaller chunks like the image below shows. 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I would be nice if that email contained a link to my inbox, which could be shown in MailWasher.  In order to collect that email in a normal email client, I'd also have to collect or delete all the other emails sent to that address.  I'm not entirely clear what's been done in that email to stop MailWasher seeing the link.  I can see a link to leave feedback, and one to unsubscribe, but not one to check my inbox.  Sure, I can live without one, but those emails would be a lot more useful to me if I could just click a link to go to my Inbox. "Its" as a Possessive Pronoun Since its can be both determiner possessive pronoun and nominal possessive pronoun, an example of its as determiner possessive pronoun would be:     We saved this question for last because of its complexity.   Because complexity is a noun, so its must be a determiner possessive pronoun in this sentence. I'm looking for an example where its is used as nominal possessive pronoun in a sentence as above. Round a double to x significant figures Reputation and inbox notification shows up again after refresh This was asked before too. .  Currently I am facing the issue of appearance of reputation notification in above toolbar.  I can see the reputation notification, for every page refresh even after clicking and viewing the notification.  current browser: Chrome Version 36.0.1985.143 m How to make table split in two pages I am writing a paper using the elsarticle document class. However, the article contains a long table which is sent to the next page, in the middle of the page although I have specified the position as 'here'. Is there any way to make the table split if one page is not long enough for the table? Thank you very much for your very kind help. glob() - sort array of files by last modified datetime stamp $\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$ I have to show that the ring $\mathbb{Z}[X]/(X^2-1)$ is not isomorphic with $\mathbb{Z}\times \mathbb{Z}$.    I know that $(\mathbb{Z}\times\mathbb{Z})^*=\{(\pm1,\pm1)\}$, so I thought I should be looking for elements which have inverses in $\mathbb{Z}[X]/(X^2-1)$ and hopefully find more or less than 4. But I didnt succeed, so I need hints. Thanks.