Let $G=\{x,y│x^4=y^4=e,xyxy^{-1}=e\}$     1.Show that $|G|\leq 16$.      2.Suppose that $|G|=16$           a.Find $Z(G)$.          b.Find a familiar group that is isomorphic to $G/〈y^2 〉$ and show that these are isomorphic.      I am having trouble with part 2b.  So far, I have  Assume $|G|=16$ then $|G|/&lt;y^2&gt; = 8$  => $|G|/&lt;y^2&gt; = |D_4|$  I'm not sure which direction to head in after this. Any feedback is appreciated.
Studying for an exam, a review question...  Given $G=\langle x,y|x^4=y^4=e,xyxy^{-1}=e\rangle$.    Show $|G|\leq16$.   For this, I want to consider that $x^3=x^{-1}$ and $y^3=y^{-1}$ based on our assumptions. I am a little lost as to how to put the second part, $xyxy^{-1}=e$ to show there are no more than 16 elements.   If $|G|=16$, find the center of the group and find a group that is isomorphic to $G/\langle y^2\rangle$.   I'm pretty sure the group is centerless.
I am trying to perform stepwise regression for variable selection in R.  In matlab, the stepwisefit function is able to work in n &lt; p problems. Trying to use step() for such problems i get the error message AIC is -infinity for this model, so 'step' cannot proceed. Is there a way to modify parameters so i can use step function? Limit Ordinal Question how to prove this result.     $\alpha$ is limit ordinal if and only if $\beta&lt;\alpha$ implies $\beta+1&lt;\alpha$ for any $\beta$.   I am getting confused because of the definition of limit ordinal. it says     $\alpha=\sup\{\beta: \beta&lt;\alpha\}=\bigcup\alpha$    for sup I am ok but for $\alpha=\bigcup\alpha$ how ?? My csv export displaying html, how to get rid of? Soft version of the maximum function? In the book Deep Learning, it says the softmax function is de facto a soft argmax function, and the corresponding soft version of the maximum function is $$\text{softmax} (z)^T z$$ How to understand the latter? Unable to correct problems, you have held broken packages After upgrading from 10.04 to 12.04 I am trying to install different packages.  For instance ia32-libs and skype (4.0).  When trying to install these, I am getting the 'Unable to correct problems, you have held broken packages' error message.   Output of commands:  sudo apt-get install -f Reading package lists... Done Building dependency tree        Reading state information... Done 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded.   After running this:  sudo dpkg --configure -a foo@foo:~$ sudo apt-get install -f Reading package lists... Done Building dependency tree        Reading state information... Done 0 upgraded, 0 newly installed, 0 to remove and 0 not upgraded. Table with multiple merging Most of the places, examples for merging of cells in tables are given. But, I could not find a merging of cells in complicated fashion as shown below. This question may summarize all the possible methods of merging and may serve as the reference for all future questions on merging of cells. Experts can add much more complications if required. If $L_i/K$ are Galois extensions of number fields, $i=1,\ldots,n$, and $L=L_1\cdots L_n$ is the compositum. Then it's true that a prime $\mathfrak{p}$ of $K$ splits in $L$ if and only if it splits in all of $L_i$. Does this also hold if the $L_i$ are not Galois extensions of $K$?  The proof that I know regarding the compositum uses the fact that the $L_i/K$ are Galois, so is this true in the more general setting and how would one prove it? I was thinking about something the other day (that is probably stupid), and thought this would be a good place to talk about it.  If someone wanted to draw its family tree over the last 1000 years, he'd write a binary tree with a depth of about 40 (counting 25 years on average between each generations). This tree would have 240 leaves, or about 1012 ancestors on the deeper level. That is obviously much more than the total world population and a stupid figure when you think about it.  The only explanation that I can think of is that a real genealogy tree must have "linked" branches at several levels that change the actual number of ancestors. Anyway, for an average person, how many ancestors would that make? Could one assume that all or nearly all of the persons that lived in its country or world region a 1000 years ago would be part of its ancestors? Factorial of 0 - a convenience? Do Hutts find humanoid females attractive, and if so why? Jabba the Hutt kept scantily clad humanoid slaves.  Jabba even tried to kiss Leia.  But IIRC Hutts are hermaphrodites and of course should by all biological reasoning not be attracted to creatures of a specific sex (don't recall sexy dudes hanging around) with radically different body design.  So do Hutts find humanoid females hot, and if so why?  Or do they just do this for status and to show off to the largely humanoid populations that surround them? $P_n(x):=1+ \sum_{m=1}^n\dfrac{x^m}{m!}$ has no real root for even $n$ and exactly one real root for odd $n$ Proof by Induction: Prove that $2^n > n^2$, for all natural numbers greater than or equal to $5$ What is the historic present tense? I learned that the historical present may be used to create an effect of immediacy in narratives. I have a question about the historic present tense. Why don't we use the progressive tense instead of the simple present tense?   ※My English skill is poor, so my question might sound rude. But I don't mean to offend anyone. I'd be glad if someone could reply to my question. Possible to do a MySQL foreign key to one of two possible tables? Well here's my problem I have three tables; regions, countries, states.  Countries can be inside of regions, states can be inside of regions.  Regions are the top of the food chain.  Now I'm adding a popular_areas table with two columns; region_id and popular_place_id.  Is it possible to make popular_place_id be a foreign key to either countries OR states.  I'm probably going to have to add a popular_place_type column to determine whether the id is describing a country or state either way. Can you explain this word to me?   Can I rephrase it as     Can you explain me this word? In the inbox of the app, there is no way to differentiate between comments and answers as is in the global inbox of the website.  Could this be added? Elegant way of obtaining the envelope of oscillating function I'm working on the question     Let $G$ be the group of invertible real matrices of the form $\left [\begin{array}{c c}a &amp; b\\ &amp; a^2\end{array}\right ]$. Determine the Lie algebra $L$ of $G$, and compute the bracket on $L$.   I'm familiar with how to derive the Lie algebra for a linear group like $U_n$, $SU_n$, etc. but I'm not sure what to do in a more explicit case like this. How do you express that your understanding of something is completely changed? When something blows your mind and changes your understanding of something that you've believed in for years, how do you express it (formally or informally)? I need to create a metadata-table for all files in a File Geodatabase.   Is it possible to automatically be updated with new filenames, and possibly even metadata (as created in the ArcGIS metadata editor)?