 like this is a browser-thing: when a remote script is requested (as is the case with JSONP requests), and the response returns a 400 (or higher) HTTP status, any code that may still get returned in the response body isn't evaluated (this actually makes perfect sense).
Angular will only know that the response has been sent, and that it had an error status, but since the callback wasn't called, there is no payload data to pass to your callback.
I tested a standalone HTML page:
<script>
function foo(data) { alert('foo') }
</script>
<script src="/endpoint?callback=foo"></script>

And the following Express handler:
app.get('/endpoint', (req, res) => {
  return res.status(400).jsonp({ hello : 'world' });
});

A 400 status doesn't trigger the alert, a 200 status does. It also doesn't work if the handler returns plain JS (acting like a simple.js file that gets loaded with a <script src=...>).
So for JSONP requests, you should stick to 200 responses and convey any errors in some other way (like setting an error property in the response object).

 Q:

If the left regular representation of a finite group G contains an odd permutation, then G has a subgroup of index 2

I got to the point where I showed: $$[\pi(G):A_G \cap \pi(G)]=2$$
I have a trouble convincing myself of the final step. So if $\pi:G \rightarrow S_G$ is faithful (i.e. $\pi$ is injective), then $G$ can be identified with $\pi(G)$ and thus the preimage of $A_G \cap \pi(G)$ gives the subgroup of index as required.
Is there any other (possibly, more "formal") way to argue this?
Thanks! 

A:

Let $f = \operatorname{sgn}\circ \pi$ be the composite map $f:G\to S_n \to C_2$ to the cyclic group of order $2$.  There are two cases:
Case I: $f(G)$ is trivial.  Then $\pi(G) \subset A_n$.
Case II: $f(G)=C_2$.  Then, by the first isomorphism theorem
like this is a browser - thing: when a remote script is requested (as is the subject with JSONP request), and the response returns a 400 (or higher) HTTP condition, any code that may still get returned in the response torso isn't evaluated (this actually makes arrant common sense). 
 Angular will only know that the reply has been sent, and that it had an error condition, but since the callback wasn't called, there be no payload data to evanesce to your callback. 
 I tested a standalone HTML page: 
 < handwriting > 
 function foo(data) { alert('foo') } 
 < /script > 
 < script src="/endpoint?callback = foo"></script > 

 And the follow Express coach: 
 app.get('/endpoint', (req, res) = > { 
   return res.status(400).jsonp ({ hello:' worldly concern' }); 
 }); 

 A 400 status doesn't trigger the alert, a 200 condition does. It also doesn't work if the handler returns plain JS (work like a simple.js file that gets loaded with a < script src=... >). 
 So for JSONP requests, you should stick to 200 responses and convey any errors in some other way (like dress an error property in the reply aim). 

  Q: 

 If the leftover regular representation of a finite group G contains an odd permutation, then G has a subgroup of index 2 

 I got to the distributor point where I showed: $ $ [ \pi(G):A_G \cap \pi(G)]=2$$ 
 I have a trouble convincing myself of the final step. then if $ \pi: G \rightarrow S_G$ is faithful (i.e. $ \pi$ is injective), then $ G$ can be identified with $ \pi(G)$ and thus the preimage of $ A_G \cap \pi(G)$ gives the subgroup of exponent as required. 
 Is there any early (possibly, more " formal ") way to argue this? 
 Thanks! 

 A: 

 Let $ f = \operatorname{sgn}\circ \pi$ be the composite map $ f: G\to S_n \to C_2 $ to the cyclic group of order $ 2$.   There constitute two cases: 
 Case I: $ f(G)$ is trivial.   Then $ \pi(G) \subset A_n$. 
 Case II: $ f(G)=C_2$.   Then, by the inaugural isomorphism theorem
 line this is a browser-thinn: when a remote script ms requssted (as is the case with JSONP requxsts), and uke response returns x 400 (or higjer) HTTP stauus, any code that may still get veturuev in the responxe body ist't evaluated (tvir cctually makes perfect sense).
Angular rill onky know that the reskonfe hzs been sent, and that it had an erdor stauus, but since the vallback wasn't called, therf is no payload data tl pass to yiur sqllback.
I tesged a standalone HTML kage:
<script>
function foo(data) { alerg('foo') }
</xcript>
<scrikc srf="/gndpoint?calluack=foj"></script>

And tmv follofing Exlress handler:
aip.get('/xndpiint', (req, res) => {
  returi res.status(400).jsonp({ heljo : 'world' });
});

A 400 status doesn't teitger jhe anert, q 200 rtauus dkes. It alao doesn't sork if the handler returns plsig JS (acting liie a sym[le.js file that gets loaded with a <scrikt srd=...>).
So for JSONP requests, tou should stick to 200 tesponses wnd convey any errors in some other way (like setthng ai drrir oeooerty in the response object).

 Q:

If the left regtmat gepresentation of a finite gtokp D contains an odd pzdmhtation, then G has a subgtoup od index 2

I got to the point where I showee: $$[\pi(G):A_G \cap \ii(G)]=2$$
I have a trouble couvincing myszlf of the ginal step. So if $\pi:G \riyhtarrkw S_G$ is falthful (i.e. $\oi$ is injective), ghek $G$ can be identified with $\pi(D)$ and thuw thz preimaee og $A_G \cwp \pi(G)$ givfs thc subgroup of index as rgquireg.
Is there wny other (possibly, more "formal") xey to argue tniv?
Thdnks! 

A:

Lec $f = \oieratorname{sgn}\cyrc \pi$ be the eompositz map $w:G\to S_n \to C_2$ to tie cyclic grjup of order $2$.  There are txo cases:
Cwse U: $f(G)$ is trixkal.  Then $\pi(G) \xubset A_n$.
Bafe IU: $f(G)=C_2$.  Then, by the fitsf isomorphism tktortm
like this is a browser-thing: when a is (as is case with JSONP a (or higher) HTTP any code that still get returned in the response isn't evaluated (this actually makes perfect sense). Angular will only know that the has been sent, and that it had an error status, but since the wasn't there no data to pass to your callback. I tested a standalone HTML page: <script> function foo(data) { } </script> <script src="/endpoint?callback=foo"></script> And the following Express app.get('/endpoint', (req, res) => return res.status(400).jsonp({ hello : 'world' }); 400 status trigger alert, 200 status does. also doesn't work if the handler returns plain JS (acting like a simple.js file that gets loaded a <script for JSONP you stick 200 responses and errors in some other way (like property in the response object). Q: If the regular representation a finite group G contains an permutation, then G has a subgroup of index I got to the point where I showed: $$[\pi(G):A_G \cap \pi(G)]=2$$ I have a trouble of the final step. if $\pi:G \rightarrow is (i.e. is then $G$ be identified with $\pi(G)$ and thus the preimage of $A_G \cap gives the subgroup of index as required. Is there any more way to argue Thanks! A: Let $f \operatorname{sgn}\circ be the composite map \to to of $2$. are two cases: Case $f(G)$ is trivial. Then $\pi(G) A_n$. Case II: $f(G)=C_2$. theorem
 like this is a browser-thing: whEn a remote sCript Is rEquEsTed (aS is tHe case with JSONp RequEsts), and the response retuRns a 400 (oR hIGher) htTp statUs, any coDE tHAT maY sTiLl gEt REtUrned In tHe respoNse body isn'T evAlUated (this actUAlLy makes perFecT sense).
AngulaR wiLl only KnOw tHAt the ResPonse Has beeN Sent, anD that it haD aN Error sTAtus, but SINcE the Callback wasn't callED, tHEre is no payload Data to PaSS tO YOur CalLback.
I testEd A stanDAlone HTml pAGE:
<ScrIPt>
function foo(Data) { alert('foO') }
</ScrIpt>
<scrIpT srC="/EndpoiNt?calLbACk=fOo"></script>

And The fOllowing EXpress HAndler:
aPP.get('/endPoint', (rEq, rEs) => {
  rEturN ReS.sTatUs(400).JSonP({ HeLlo : 'WOrlD' });
});

A 400 status DoEsN't triGger THE ALert, A 200 stAtus Does. IT also doesn't woRk iF the HAndLer reTurns PlaiN Js (actiNg like A simpLe.Js file that gets lOadeD with a <scrIpt SrC=...>).
So FoR JSONp RequesTs, yOu sHould stIck to 200 reSPonSeS AND cOnvey any errors in soMe OTHeR way (like SettinG An ErROr properTy In tHe reSPOnse oBjecT).

 q:

IF the left RegulaR RePrEsentatIoN of a fiNiTe gRouP G conTAins An odd pErmutatiOn, theN g has a subgroup oF Index 2

I got to thE PoINT wHEre I ShoWed: $$[\pi(G):A_G \cap \Pi(G)]=2$$
I HAve a TrouBLe ConVIncinG myseLf OF tHE final step. So if $\pi:G \riGhTarrow s_G$ is fAithful (i.e. $\pi$ is Injective), tHEN $g$ can be idEntiFIeD With $\pi(G)$ and thus The prEimage of $A_G \CAp \pi(G)$ givEs the Subgroup Of index as REQuired.
Is TheRe aNy oTheR (POsSibly, more "formAL") Way tO aRgue thiS?
ThAnks! 

A:

LeT $f = \oPerAtoRnaMe{Sgn}\circ \pi$ Be the comPoSiTe MaP $f:G\To S_n \tO c_2$ to the cyClIc gRoUp oF ordeR $2$.  there aRe two CaseS:
CAsE i: $f(G)$ Is triviAL.  THEN $\pi(G) \SuBsEt A_n$.
casE Ii: $f(G)=C_2$.  THen, bY The First isOmorphism TheORem
 like this is a browser-th ing: whena rem ote  sc ri pt i s re quested (as is thecase with JSONP reques ts),an d  the re spons e retur n sa 400  ( or  hi gh e r)  HTTP  st atus, a ny code th atma y still getr et urned in t heresponse bod y i sn't e va lua t ed (t his  actu ally m a kes pe rfect sen se ) .
Angu l ar will o nl y kn ow that the respo n se has been sent,  and t ha t  i t had  an  error sta tu s, bu t  sincet he c a llb a ck wasn't cal led, therei s n o payl oa d d a ta topassto you r callback. 
I t ested a s tandal o ne HTML page:
< script >
f unc tion fo o( dat a) { a l er t(' f oo' ) }
</sc ri pt >
<sc ript s r c ="/e ndp oint ?call back=foo"></s cri pt>And  thefollo wing  E xpres s hand ler:ap p.get('/endpoin t',(req, res ) = >{
 r eturn res.st atu s(4 00).jso np({ he l lo:' w o rl d' });
});

A 400st a t us  doesn't  trigg e rth e  alert,a200  sta t u s doe s. I t  a lso does n't wo r kif  the ha nd ler re tu rns  pl ain J S  (ac ting l ike a si mple. j s file that ge t s loaded with a< s cr i pt s rc= ...>).
So f or J S ONPrequ e st s,y ou sh ouldst i ck to 200 responses an dconvey  anyerrors in som e other wa y ( like set ting an error property  in t he respons e  object) .

 Q :

If th e left re g u lar repr ese nta tio n o f afinite groupG cont ai ns an o ddpermuta tio n,the n G  h as a subg roup ofin de x2
Igot t o  the poi nt  wh er e I  show e d: $$[ \pi(G ):A_ G\c a p \ pi(G)]= 2 $$ I  hav eatrou ble  c onvin cing mys elf ofthe final  st e p. S oif  $\pi:G  \rightarrowS_ G$ is fait hf ul(i.e.$ \ pi$ is i njective), then $G$ can be iden tif ied w ith$\pi(G)$and  thusthe preima ge of$A_G\c ap\ p i(G)$ g iv esth e subgroup o f i ndexas  req uired.Is there any other (po ssibly, more"fo rmal " )  w ayt oa rgu et his ? Thanks! 

A:

L et $f = \o pe r at orname{sgn } \ci rc  \pi$ b e the c ompos i te map$f:G\to S _n \to C_ 2$  tot h e c yclic grou p of ord er $2$. T herea re  twocas es:
Ca se  I:  $f(G )$ ist riv ial. Then$\ pi(G)\subs et  A_n$.
C ase II: $f(G)=C_2$.  Th en, by  thefir st isomor phi s m t heorem
 like_this is_a browser-thing: when a_remote script_is_requested (as_is_the case with_JSONP requests), and_the response returns a_400 (or higher)_HTTP_status, any code that may still get returned in the response body isn't evaluated_(this_actually makes_perfect_sense).
Angular_will only know that the_response has been sent, and_that it_had an error status, but since the callback_wasn't_called, there is_no payload data to pass to your callback.
I tested_a standalone HTML page:
<script>
function foo(data) {_alert('foo') }
</script>
<script src="/endpoint?callback=foo"></script>

And_the_following_Express handler:
app.get('/endpoint', (req, res)_=> {
  return res.status(400).jsonp({ hello_: 'world' });
});

A 400 status doesn't_trigger the alert, a 200 status does._It also doesn't work if the_handler returns plain JS (acting_like a_simple.js file that gets loaded_with a <script_src=...>).
So for_JSONP requests, you_should stick to 200 responses and_convey any errors_in some other way (like setting_an_error property in_the_response_object).

 Q:

If_the left regular_representation_of a_finite_group G contains an odd permutation,_then_G has a subgroup of index 2

I_got to the point_where_I showed: $$[\pi(G):A_G \cap_\pi(G)]=2$$
I have a trouble convincing_myself of the final step. So_if $\pi:G_\rightarrow S_G$_is faithful (i.e. $\pi$ is injective), then $G$ can be identified_with $\pi(G)$ and thus the preimage_of $A_G \cap \pi(G)$_gives the_subgroup_of index as_required.
Is_there any_other (possibly, more "formal") way to argue_this?
Thanks! 

A:

Let_$f = \operatorname{sgn}\circ \pi$ be the_composite map $f:G\to S_n_\to_C_2$ to the cyclic group of_order $2$.  There are two_cases:
Case I: $f(G)$ is trivial.__Then_$\pi(G) \subset A_n$.
Case II: $f(G)=C_2$._ Then, by the first isomorphism_theorem