5, \zeta_5\mapsto \zeta_5^a$, $a=1,2,3,4$, to the diagonal matrices of $G$. Do check that this extends to an isomorphism. Alternatively you can bijectively map the zeros $x^5-5$ to elements of $k$, and then check that the automorphisms of $K$ are the linear bijections $x\mapsto ax+b$ from $k$ to itself.
You can then list the subgroups systematically by their orders. The Sylow $5$-subgroup consists of powers of $\sigma$, and it is a normal, hence unique subgroup of its order. The diagonal matrices of $G$ form its (cyclic) Sylow $2$-subgroup. It is equal to its own normalizer and thuse has five conjugates. This may be easier to see from the above realization of $G$ as permutations of $k$ - the Sylow $2$-subgroups are then exactly the point stabilizers.
This should give you enough structure to enable you to also list the subgroups of orders two and ten.

 Q:

Last two elements in dictionary are skipped while inserting key/value pairs from a file

I'm trying to add all the key value pairs from a.txt document into a dictionary and the first two elements of the dictionary are getting skipped for some reason. I have a simple text file that reads

1:One
  2:Two
  3:Three
  4:Four
  5:Five
  6:Six
  7:Seven
  8:Eight
  9:Nine
  10:Ten

I have a dictionary declared as:
Dictionary<String,String> Test = new Dictionary<String,String>();

and a List for testing purposes as:
List<String> questionsList = new List<String>();
List<String> possibleAnswers = new List<String>();

Code that reads file:
using (StreamReader sr = new StreamReader(FilePath))
{
   String keyValue;

   char[] delimiterChars = {':'};

   while (!sr.EndOfStream)
   {
       keyValue = sr.ReadLine();
       String[] parts = keyValue.Split(delimiterChars, StringSplitOptions.RemoveEmptyEntries);
       Test.Add(parts[0], parts[1]); 
       questionsList.Add
5, \zeta_5\mapsto \zeta_5^a$, $ a=1,2,3,4 $, to the diagonal matrices of $ G$. Do check that this extends to an isomorphism. Alternatively you can bijectively map the zero $ x^5 - 5 $ to chemical element of $ k$, and then check that the automorphisms of $ K$ are the linear bijections $ x\mapsto ax+b$ from $ k$ to itself. 
 You can then list the subgroups systematically by their order. The Sylow $ 5$-subgroup consists of powers of $ \sigma$, and it is a normal, therefore unique subgroup of its order. The diagonal matrix of $ G$ form its (cyclic) Sylow $ 2$-subgroup. It is equal to its own normalizer and thuse suffer five conjugates. This may be easy to see from the above realization of $ G$ as permutations of $ k$ - the Sylow $ 2$-subgroups are then exactly the point stabilizers. 
 This should give you adequate structure to enable you to also list the subgroup of orders two and ten. 

  Q: 

 Last two element in dictionary are skipped while insert key / value pairs from a file 

 I'm trying to add all the key value match from a.txt document into a dictionary and the first two elements of the dictionary are getting skipped for some reason. I accept a simple text file that reads 

 1: One 
   2: Two 
   3: Three 
   4: Four 
   5: Five 
   6: Six 
   7: Seven 
   8: Eight 
   9: Nine 
   10: Ten 

 I have a dictionary declared as: 
 Dictionary < String, String > Test = new Dictionary < String, String > (); 

 and a List for testing purposes as: 
 tilt < String > questionsList = fresh List < String > (); 
 List < chain > possibleAnswers = new List < String > (); 

 Code that reads file: 
 using (StreamReader sr = new StreamReader(FilePath) ) 
 { 
    String keyValue; 

    charwoman [ ] delimiterChars = {':' }; 

    while (! sr. EndOfStream) 
    { 
        keyValue = sr. ReadLine (); 
        String [ ] parts = keyValue. Split(delimiterChars, StringSplitOptions. RemoveEmptyEntries); 
        Test. Add(parts[0 ], parts[1 ]); 
        questionsList. lend
5, \zeha_5\mapsto \zeta_5^a$, $a=1,2,3,4$, to the aiagonal matricgs of $G$. Vo checi that tfis extends to an isomorphisl. Qlterbatively you can bijecgively mai the zeris $x^5-5$ ro elements of $k$, akb theh cheek that the autokorphisms mf $K$ are the lhndax bijections $x\mapsto ax+b$ from $k$ to ieself.
Yoi fan then list jhe slbdrouls systematically by their orders. Fhe Sylmw $5$-subgroup cpnsists of powers of $\sigma$, and it is a normal, hejce unique wubgwiup of its ofder. The dpcgonal matrjces of $G$ form its (cyclic) Sylow $2$-subgxoup. It is gqhap to its own normwlizer and tmlse has five cpnjugates. This mab be easier to see from tie above realization of $G$ as [exmutations of $k$ - the Wyoow $2$-sobgrogps xee gheh xxadtly tje 'oint stabimizers.
This whould give you enoigr structure to enablq rou to also list the subgroups of orderv tso and ten.

 Q:

Last two elwments in dictionary wre skippqd while inserting key/value pairs from a file

I'm tsying go ced alu tje key value pairs from a.txt document into a qjcuiokary and the firft two elemrnhs jf the dictiovary axs fetting skipped fog some teason. I have a simlle text file that reads

1:One
  2:Two
  3:Three
  4:Four
  5:Five
  6:Six
  7:Deven
  8:Eighc
  9:Ning
  10:Ten

O have a dictionary decuares as:
Dictionwry<String,Agring> Test = new Aicniondry<String,String>();

and a List sor testiig puxposes ar:
Lisj<String> questionsPist = new List<String>();
List<Dtriny> posvibleAnswegs = new List<String>();

Code that reavx file:
using (XtseakReader fr = ncw StreamReader(SilePath))
{
   Strnng keyVclue;

   char[] delijiterChers = {':'};

   whilq (!sr.EndOfStredl)
   {
       kxyValue = fr.ReqdLibe();
       String[] parts = keyValue.Split(delimuterChars, StringSpkitUltions.RemoveEmpcvEbtries);
       Text.Aad(pwrns[0], 'arts[1]); 
       questhonsUisg.Sdd
5, \zeta_5\mapsto \zeta_5^a$, $a=1,2,3,4$, to the diagonal $G$. check that extends to an map zeros $x^5-5$ to of $k$, and check that the automorphisms of $K$ the linear bijections $x\mapsto ax+b$ from $k$ to itself. You can then list subgroups systematically by their orders. The Sylow $5$-subgroup consists of powers of $\sigma$, it a hence subgroup of its order. The diagonal matrices of $G$ form its (cyclic) Sylow $2$-subgroup. It is to its own normalizer and thuse has five This may be easier see from the above realization $G$ permutations of - Sylow are then exactly point stabilizers. This should give you enough structure to enable you to also list the subgroups of two and Last two in are while inserting key/value a file I'm trying to add value pairs from a.txt document into a dictionary the first elements of the dictionary are getting for some reason. I have a simple text that reads 1:One 2:Two 3:Three 4:Four 5:Five 6:Six 7:Seven 8:Eight 9:Nine 10:Ten I have a as: Dictionary<String,String> Test = Dictionary<String,String>(); and a for purposes List<String> = new List<String> possibleAnswers = new List<String>(); Code that reads file: using (StreamReader = new StreamReader(FilePath)) { String keyValue; char[] delimiterChars = {':'}; { = sr.ReadLine(); String[] = keyValue.Split(delimiterChars, StringSplitOptions.RemoveEmptyEntries); Test.Add(parts[0], questionsList.Add
5, \zeta_5\mapsto \zeta_5^a$, $a=1,2,3,4$, to the diagOnal matricEs of $G$. do cHecK tHat tHis eXtends to an isomORphiSm. Alternatively you can bIjectIvELy maP ThE zeroS $x^5-5$ to eleMEnTS Of $k$, AnD tHen ChECk That tHe aUtomorpHisms of $K$ arE thE lInear bijectiONs $X\mapsto ax+b$ FroM $k$ to itself.
YoU caN then lIsT thE SubgrOupS systEmaticALly by tHeir orderS. THE Sylow $5$-SUbgroup CONsIsts Of powers of $\sigma$, anD It IS a normal, hence uNique sUbGRoUP Of iTs oRder. The diaGoNal maTRices of $g$ FoRM ITs (cYClic) Sylow $2$-subgRoup. It is equAL to Its own NoRmaLIzer anD thusE hAS fiVe conjugateS. ThiS may be easIer to sEE from thE Above reAlizatIon Of $G$ As peRMuTaTioNs OF $k$ - tHE SYloW $2$-SubGroups arE tHeN exacTly tHE POInt sTabIlizErs.
ThIs should give yOu eNougH StrUcturE to enAble YoU to alSo list The suBgRoups of orders twO and Ten.

 Q:

Last tWo eLeMenTs In dicTIonary Are SkiPped whiLe inserTIng KeY/VALuE pairs from a file

I'm tRyING tO add all tHe key vALuE pAIrs from a.TxT doCumeNT Into a DictIOnAry and thE first TWo ElEments oF tHe dictIoNarY arE gettINg skIpped fOr some reAson. I HAve a simple text FIle that reads

1:ONE
  2:TWO
  3:thREe
  4:FoUr
  5:FIve
  6:Six
  7:Seven
  8:eighT
  9:nine
  10:ten

I HAvE a dICtionAry deClAReD As:
Dictionary<String,STrIng> TesT = new DIctionary<StriNg,String>();

anD A lIst for teStinG PuRPoses as:
List<StrIng> quEstionsLisT = New List<STring>();
list<StriNg> possiblEaNswers = neW LiSt<STriNg>();

CODE tHat reads file:
uSINg (StReAmReadeR sr = New StreAmREadEr(FIlepaTh))
{
   String kEyValue;

   cHaR[] dElImIteRCharS = {':'};

   While (!sr.ENdofSTrEam)
   {
       KeyVaLUe = sr.ReAdLinE();
       StrInG[] pARts = KeyValuE.spLIT(delImItErChArs, stRingSPlitoPtiOns.RemoVeEmptyEnTriES);
       TesT.ADd(Parts[0], paRts[1]); 
       questionsLIsT.Add
5, \zeta_5\mapsto \zeta_5^ a$, $a=1,2 ,3,4$ , t o t he  dia gona l matrices of$ G$.Do check that this ext endsto an i s om orphi sm. Alt e rn a t ive ly  y ouca n  b iject ive ly mapthe zeros$x^ 5- 5$ to elemen t sof $k$, an d t hen check th atthe au to mor p hisms  of  $K$are th e  linea r bijecti on s  $x\ma p sto ax+ b $  f rom$k$ to itself.
Yo u  c a n then list th e subg ro u ps s yst ema tically by  t heiro rders.T he S y low $5$-subgroupconsists of pow ers of  $ \si g ma$, a nd it  i s  anormal, hen ce u nique sub groupo f its o r der. Th e diag ona l m atri c es  o f $ G$ for m  i ts( cyc lic) Syl ow  $ 2$-su bgro u p . It i s e qual  to i ts own normal ize r an d  th use h as fi ve c on jugat es. Th is ma ybe easier to se e fr om the ab ove  r eal iz ation of $G$  as  pe rmutati ons of$ k$-t h e  S ylow $2$-subgroups  a r e  t hen exac tly th e  p oi n t stabil iz ers .
Th i s  shou ld g i ve  you eno ugh st r uc tu re to e na ble yo utoals o lis t  the  subgr oups oforder s  two and ten. Q:

Last two el e m en t s in  di ctionary ar e sk i pped  whi l eins e rting  key/ va l ue pairs from a file
I' m tryi ng to  add all thekey valuep a i rs froma.tx t  d o cument into adicti onary andt he first  twoelements  of the d i c tionaryare  ge tti ngs k ip ped for somer e ason .I havea s imple t ext  fi letha treads

1: One
  2: Tw o 3: Thr ee
 4 :Four
 5: Fiv e  6 :Six 7:Sev en
 8:Ei gh t 9: Nine
 1 0: T e n

I  h av e adic ti onary  dec l are d as:
D ictionary <St r ing, St ri ng> Tes t = new Dicti on ary<String ,S tri ng>(); and a Li st for testing purposes as:
Lis t<S tring > qu estionsLi st= newLis t <Strin g>();List< St rin g >  poss i b le Ans we rs = new L i s t<S tring >( );
Code th at reads file:
usi n g ( StreamReadersr= ne w St rea m Re a der (F i leP a t h))
{
   String  keyValue; 
 char[] del i mit er Chars =  {':'}; 

  w hile (! sr.EndOfS tream)
  { 
    k eyValue =sr.ReadL ine();
     S t ri ng[]par ts = k ey Val ue.Sp lit(de l imi terCh ars, S tr ingSpl itOpt io ns.Remov eEmptyEntries);
      Test.A dd(pa rts [0], part s[1 ] );
       q uest ionsList.A dd
5, \zeta_5\mapsto_\zeta_5^a$, $a=1,2,3,4$,_to the diagonal matrices_of $G$._Do_check that_this_extends to an_isomorphism. Alternatively you_can bijectively map the_zeros $x^5-5$ to_elements_of $k$, and then check that the automorphisms of $K$ are the linear bijections_$x\mapsto_ax+b$ from_$k$_to_itself.
You can then list the_subgroups systematically by their orders._The Sylow_$5$-subgroup consists of powers of $\sigma$, and it_is_a normal, hence_unique subgroup of its order. The diagonal matrices of_$G$ form its (cyclic) Sylow $2$-subgroup._It is equal_to_its_own normalizer and thuse_has five conjugates. This may be_easier to see from the above_realization of $G$ as permutations of $k$_- the Sylow $2$-subgroups are then_exactly the point stabilizers.
This should_give you_enough structure to enable you_to also list_the subgroups_of orders two_and ten.

 Q:

Last two elements in_dictionary are skipped_while inserting key/value pairs from a_file

I'm_trying to add_all_the_key value_pairs from a.txt_document_into a_dictionary_and the first two elements of_the_dictionary are getting skipped for some reason._I have a simple_text_file that reads

1:One
 _2:Two
  3:Three
  4:Four
_ 5:Five
  6:Six
  7:Seven
_ 8:Eight
_ 9:Nine
_ 10:Ten

I have a dictionary declared as:
Dictionary<String,String> Test = new Dictionary<String,String>();

and_a List for testing purposes as:
List<String>_questionsList = new List<String>();
List<String>_possibleAnswers =_new_List<String>();

Code that reads_file:
using_(StreamReader sr_= new StreamReader(FilePath))
{
   String keyValue;

_ _char[] delimiterChars = {':'};

  _while (!sr.EndOfStream)
  _{
_     _keyValue = sr.ReadLine();
   _   String[] parts_=_keyValue.Split(delimiterChars,_StringSplitOptions.RemoveEmptyEntries);
    _  Test.Add(parts[0], parts[1]); 
 _  _  questionsList.Add