Question is to prove that :  $A_n$ contains an isomorphic  copy of $S_{n-2}$  I have no idea from where to start.  I do not even believe this can be true (expect for cardinality conditions).  But as this is an exercise.it has to be true.   please provide some hints/suggestions to attack this problem.  Thank you :)
I'm working through Rotman (1994). It's the 2nd chapter where my curiosity lead me to a question that is not from the book. Here's the problem that inspired the question:      2.8. Imbed $S_n$ as a subgroup of $A_{n+2}$, but show, for $n\ge2$, that $S_n$ cannot be imbedded in $A_{n+1}.$    Since the alternate subgroup has the property of even parity, I thought that a little investigation of parity for the second and third symmetric groups was in order. So I constructed a table for each of them, grouped by even and odd number of disjoint products.   I'll forego writing the actual table and simply provide the answers. In symmetric groups with an even idegree, two even permutations come together under composition to produce an even permutation. Two odds also make an even. Two different parities produce odd elements.   When the symmetric group has an odd degree, e.g. $S_3$, two odds make an odd, two evens make an odd, and two different parities make an even element. Exactly the opposite results from the symmetric groups with even index.   Two questions: is this enough to prove the $A_{n+1}$ case? And can this be generalized modulo 2 e.g. $A_{2n+1}$?
How can I show that the set of rational numbers with denominator a power of two form a dense subset of the reals? How can I show that the set of rational numbers with denominator a power of two form a dense subset of the reals? Prove that the number of $\alpha\in\mathbb{F}_{27}$ such that $|A_\alpha|=26 $ equals 12. Let $\mathbb{F}_{27}$ denote the finite field of size 27. For each $\alpha\in\mathbb{F}_{27}$ , define $ A_{\alpha}$ $=$ { $ 1, 1+\alpha , 1+\alpha+{\alpha}^2 , 1+\alpha+{\alpha}^2+{\alpha}^3 , .... $} . Then  prove that  the number of  $\alpha\in\mathbb{F}_{27}$ such that $|A_\alpha|$=26 equals 12.  $ 0\in A_\alpha $ if and only if $\alpha\neq 0$ For the first proof , the multiplicative group of the field $\mathbb{F}_{27}$ is cyclic and its order is 26  , so there are 12 generators in that group.   Now  I need  to  show  that  $ 1+\alpha+{\alpha}^2+.. {\alpha}^{25} $ =0  , where order of $\alpha$=26 . Since   $\alpha ^{26}$ =1 ($\alpha$-1)(1+$\alpha$+${\alpha}^2$+...+${\alpha}^{25}$)=0 , since $\alpha\neq 1$  , So $ 1+\alpha+...+{\alpha}^{25} $ =0 My question is , can I write these last two  lines?  I mean are those two lines alright to write? If those two lines are invalid then I need some help to show  $ 1+\alpha+{\alpha}^2+..+{\alpha}^{25} $=0  .Thanks The following question is on my homework assignment that I cannot figure out:     Let U and V be independent random variables, each having a normal distribution with mean zero and variance one. Find the moment generating function of the random variable W = UV .   I have looked around online, and cannot find an answer to this question. In fact, the only answers I can find that even relate to the product of standard normal random variables are using techniques that we never covered in my class. We covered in class how to find the MGF for linear combinations of random variables, by W isn't linear, its a product of two normals. So that technique won't work.   What am I supposed to do? I am completely at a loss. I have tried multiplying the MGFs of U and V together, but that leaves me with something ugly that I can't reduce. Sum data based on cell color in Google Spreadsheets I have seen  and altough I face the same problem, the answer to that question is not helpful to my case.   I have large spreadsheets which are changed frequently. So, according to the solution accepted to that question, it is hard to duplicate all populated columns and it is cumbersome to edit one more cell everytime I wish to make a change on cells background colors.  Isn't there a way to sum data based on a condition regarding cell background color in Google Spreadsheets? Exclude property from serialization via custom attribute (json.net) Travel to Australia with new passport (with visa in old passport) I was granted a visa to Australia few months ago. I’d like to travel to Australia in few days. However, I’ve just renewed my passport and I don’t have enough time to inform them about my passport renewal. Can I travel to Australia by carrying both my old and my new passports? Query a region within a larger region with GeoDjango REST API I have a working Django REST API which can query all the counties in a country. //My model class kenya_counties(models.Model):   objectid = models.BigIntegerField()   id_field = models.BigIntegerField()   county_nam = models.CharField(max_length=80)   const_code = models.BigIntegerField()   constituen = models.CharField(max_length=80)   county_cod = models.BigIntegerField()   shape_leng = models.FloatField()   shape_area = models.FloatField()   geom = models.MultiPolygonField(srid=4326)    def __str__(self):       return self.county_nam  //My Serializer class CountiesSerializer(serializers.ModelSerializer):   class Meta:      model = kenya_counties      fields = ('objectid', 'id_field', 'county_nam', 'const_code', 'constituen', 'county_nam', 'county_cod', 'shape_leng', 'shape_area')  //My view @api_view(['GET', 'POST']) def counties(request):    if request.method == 'GET':        data = kenya_counties.objects.all()         serializer = CountiesSerializer(data, context={'request': request}, many=True)         return Response(serializer.data)     elif request.method == 'POST':        serializer = CountiesSerializer(data=request.data)        if serializer.is_valid():            serializer.save()            return Response(status=status.HTTP_201_CREATED)                 return Response(serializer.errors, status=status.HTTP_400_BAD_REQUEST)  //My url     re_path(r'^api/gis/counties/$', views.counties),  I get . How can I query just one county and display it on a react-leaflet map as shown below? &lt;MapContainer center={position} zoom={13} scrollWheelZoom={false}&gt;     &lt;TileLayer       attribution='&amp;copy; &lt;a href=&quot;http://osm.org/copyright&quot;&gt;OpenStreetMap&lt;/a&gt; contributors'       url=&quot;https://{s}.tile.openstreetmap.org/{z}/{x}/{y}.png&quot;     /&gt; &lt;/MapContainer&gt; Cryptoclass Throws Error. Invalid private key. Must be 16 bytes. Sending 16 characters long key key__c :- abcdefghijklmnop  public static String decrypt(String data) {     EncryptionSettings__c settings = EncryptionSettings__c.getOrgDefaults();     if (settings.key__c == null) {         throw new FacebookException('Cannot decrypt without a key!');     }     Blob key = EncodingUtil.base64Decode(settings.key__c);     return Crypto.decryptWithManagedIV('AES128', key, EncodingUtil.base64Decode(data)).toString(); }  public static String encrypt(String data) {         EncryptionSettings__c settings = EncryptionSettings__c.getOrgDefaults();     Blob key = null;     if (settings.key__c == null) {         checkUserCanCustomizeApplication();         key = Crypto.generateAesKey(128);         settings.key__c = EncodingUtil.base64Encode(key);         insert settings;     } else {         key = EncodingUtil.base64Decode(settings.key__c);     }     return EncodingUtil.base64Encode(Crypto.encryptWithManagedIV('AES128', key, Blob.valueOf(data))); } Stop inserting hard line-wraps in blockquotes! When you use the markdown editor's blockquote feature (the quote button / Ctrl+Q), it inserts hard line-wraps in the markdown source automatically for you. That is, it transforms text like this:  Hindus regard it as an earthly manifestation of the goddess Tulsi, a consort of the god Vishnu.   The offering of its leaves is mandatory in ritualistic worship of Vishnu and his forms like Krishna and Vithoba.   Into text like this:  &gt; Hindus regard it as an earthly manifestation of the goddess Tulsi, a &gt; consort of the god Vishnu.  &gt;  &gt; The offering of its leaves is mandatory in ritualistic worship of &gt; Vishnu and his forms like Krishna and Vithoba.   The editor should instead transform it into this:  &gt; Hindus regard it as an earthly manifestation of the goddess Tulsi, a consort of the god Vishnu.  &gt;  &gt; The offering of its leaves is mandatory in ritualistic worship of Vishnu and his forms like Krishna and Vithoba.   That is, without adding any newlines.   Why on earth does it insert hard line-wraps!? Both these forms look the same when rendered, but the current way makes it more difficult for us to go back and edit the blockquoted text later if need be.   This is particularly objectionable functionality for sites that use MathJax - while the hard line-wraps don't impact the rendered output, the additional &gt; that are inserted do (they manifest as greater-than signs if they get inserted within MathJax $$...$$). See this post on Meta.Math.SE for more details: .    I found  that this behavior is apparently :      The "reflow the text when you click blockquote" is a deliberate feature, intended to improve the readability on quoted text.   Seriously? Are there any browsers that don't soft-linewrap textareas anymore? How is hard-wrapping at 70(?) characters better than soft-wrapping at $textareawidth? If readability is the concern here, then why not hard-wrap all non-code upon post submission? Every finitely generated algebra over a field is a Jacobson ring Knowing that the polynomial ring in $n$ variables over a field $k$ is a Jacobson ring, how can we prove from it that every finitely generated $k$-algebra is a Jacobson ring?   EDIT: We define a ring to be Jacobson if every prime ideal is an intersection of maximal ideals. SEO and URL Shorteners Shapefile projections not matching up in QGIS? To start off with, I'm very new to QGIS. I'm attempting to place a shapefile with locations for 3 substations on a map of Peniche, Portugal. The locations for the substations are given in latitude and longitude coordinates.   When I start off by importing the substation shapefile, I make sure that both the layer CRS and project CRS are set to EPSG:32629, which gives each substation point the correct coordinates, however the scale is at 1:1, which is not correct. As soon as I then add a raster map of the region or a shapefile of Portugal using the same CRS the position of the substations also becomes incorrect.   I'm sure there's a simple solution I'm not getting, but I've been trying for a while and have had no success. Here's a dropbox link to the files Why do we need sigma-algebras to define probability spaces? We have a random experiment with different outcomes forming the sample space $\Omega,$ on which we look with interest at certain patterns, called events $\mathscr{F}.$ Sigma-algebras (or sigma-fields) are made up of events to which a probability measure $\mathbb{P}$ can be assigned. Certain properties are fulfilled, including the inclusion of the null set $\varnothing$ and the entire sample space, and an algebra that describes unions and intersections with Venn diagrams.  Probability is defined as a function between the $\sigma$-algebra and the interval $[0,1]$.  Altogether, the triple $(\Omega, \mathscr{F}, \mathbb{P})$ forms a probability space.  Could someone explain in plain English why the probability edifice would collapse if we didn't have a $\sigma$-algebra? They are just wedged in the middle with that impossibly calligraphic "F". I trust they are necessary; I see that an event is different from an outcome, but what would go awry without a $\sigma$-algebras?     The question is: In what type of probability problems the definition of a probability space including a $\sigma$-algebra becomes a necessity?       provides a plain English accessible explanation. The idea is a spinning pointer rotating counterclockwise on a circle of unit perimeter:       We begin by constructing a spinner, which consists of a circle of unit   circumference and a pointer as shown in [the] Figure. We pick a point   on the circle and label it $0$, and then label every other point on the   circle with the distance, say $x$, from $0$ to that point, measured   counterclockwise. The  experiment  consists  of spinning the pointer and    recording the label of the point at the tip of the pointer. We let the random   variable $X$ denote the value of this outcome.  The sample space is   clearly the interval $[0,1)$.  We would like to construct a   probability model in which each outcome is equally likely to occur. If   we proceed as we did [...] for experiments with a finite number of   possible outcomes,  then we must assign the probability $0$ to each   outcome,  since otherwise,  the  sum  of  the  probabilities,  over    all  of  the  possible  outcomes,  would not equal 1.  (In fact,   summing an uncountable number of real numbers is a tricky business;    in  particular,  in  order  for  such  a  sum  to  have  any  meaning,   at  most countably many of the summands can be different than $0$.)    However, if all of the assigned probabilities are $0$, then the sum is   $0$, not $1$, as it should be.   So if we assigned to each point any probability, and given that there is an (uncountably) infinity number of points, their sum would add up to $&gt; 1$. MWE for externalize + automatic recompilation after tikz figure modification Finding an angle within an 80-80-20 isosceles triangle How do you spawn ONLY zombies in Singleplayer? How do I calculate the resistor value for a simple LED circuit? What is a Generic Handler in asp.net and its use? How do I run two isolated instances of firefox? For testing purposes, I want to install two different instances of firefox on ubuntu - so that they don't share cookies. (I want to simultaneously login with both instances as different users to our website).  How do I do that? I do it with Firefox &amp; Opera, but I prefer Firefox (because of firebug). What are the rules of parallelism? I've been reading a lot about it since yesterday and all I encounter is "They must have the same grammatical form"? What does this exactly mean?  I know how to make parallel sentences with gerunds. The things get trickier when it comes to using modifiers.  For example:     I love using the phone that you bought me and the computer.   On the left side of the conjunction there is a noun phrase that consists of an article + noun + another modifier and on the right side there is an article + noun.  They are both noun phrases but the first one has additional modifier. Is that still considered a parallel sentence?  Another example:     I like to swim and to feel good.   Here on the right side of the conjunction there is an infinitive and on the left side there is an infinitive + complement? Is that parallel?