Mth4211521 Lecture 14 9121 20det Let G Be A Group H A Subgroupfor Eac ✓ Solved

MTH Lecture Det Let G be a group H a subgroup For eachelement a EG the left coset oft associated with a is the set at haha h EH's In additive notation atte hath htt 9 we canthinkof a H as some sort of shift of H by a Therightcoset oft associated with a is defined as Ha L ha HEH9 Reward for each a aft may not equal Ha EI Let G Zg H 244 Then the left Cosets are ,24,69 I124 21,35,.46,,7 IE ,,81,33 6 t Za 26,0 2. I 359 Note that ,246 It ,35,79 EI Let G Sss H La B I write down all the left cosees of H Recall the elements of G ave C331 43 l 33 GD L23 uz 1223 clad f I 132 EI CB LDH H B It L 233cL C12 H L L R H SHE J's B H L Lab Note CDH4 B H uh 11376 IN 123H It C1323ft iz 113239 coincidence Lemme Properties of Cosets Let G be a group and H a subgroup Let a b EG Then I a CAH 2 aH H if andonly if a EH 3 CabSH albH 4 AH bH if andonly if aEbH 5 Either aH bH or at Nbt _of 6 aH bH it andonly if a b EH 7 late4171 8 att Ha if andonly if H aHaY 9 att is a subgroup of G if andonlyif a Etl tf 1,43 Exercises 4 First suppose a Ebt Then a bhp for some h EH We show attebH and bH Eat Let X beany element of a H Then A ah for some h E H Now f ah bheh bChih since it is closed hehEH so debt Since a bhi b ahit Let y be anyelement of BH Then Y bh for some HEM Now y bh ahi h aChih Gatt where weused the fact that hith EH Since XYarearbitrary we have ate BHand BH Eat So at bt 5 Suppose att n BHI d Lets be an element in aHnbH then I ah bhz for some hihaGH a bhahi Ebt By 4 att BH 6 By 4 aH bH if andonl if aEbH Now a Ebt Eh EH a bh 7HEH b a hs b la EH at BEM 7,8 Exercises 9 By 5 the distinct coset of H partitionG Amongthem only It contains e and is a subgroup By 4 aH eH H if andonly if aEH By Lemma itemsCD thedistinct coset of H form a partition of G 100002 Ee Find the distinct cosets ofH 41,35,9 159 in G U Zz L1,337,9 B 15,1719,219 Weknow 1H 3 test 9H45H L1,35,9159 Now pick anelement not in H find its cosee say 7 Ftl L 7 21,mod, BY thelemma 7A HE471317,1948 Sothese two are the distinct coset Thur71 Lagrange's thus Let G be a finite group and H a subgroupofG Then IHI divides G Morever the number of distinct Clefts assets of ft in Cr in PI By Lemma eD ta EGe a Gat So alfgate G By E Va.BE G either at bat or aH n bit of So if a H Azt AmH are all the distinct Cosets then they must be pairwise disjointand their union is G Sothey form a partition of G By CF AiHamHELH M SO 1Gt dueto thepartition f I mCHI SO m IGI th B implies Hel11611h11 and that distinct assets of Hin G e E Reward the number of left Cosets of H in G is also called the index of Hin Gand is denotedby 1Git Y 1 Cord If G B a finite groupand H asubgroup then 1betel Core If G is a finite group the order lad ofany element a must divide Gl PI sa is a cyclic subgroupofG oforder lat By thin F l lal must dicide G EI U122 LI 3,57,,1719,219 WED1 10 the only possible ordersof an element are 1 2,5 or lo For instance we saw earlier that 131 5 Cor3 Let p be a prime Let G be a group oforder p Then G must be cyclic PI Let a be any nonidentity element Then lat By Cor 3 lal mustdivide p But the onlydivisors otp are 1 and P so at p So G La Cort Let G be a group of finite ordern Let a EG Then a ee PI Let metal By Cor 3 mln So ne mK for some integer K Now an Cam keek e Corte Fermat's little Theorems For every integer a andeveryprime p APmodp a modp PI By the division algorithm a mptr where m r EE and o Er Ep 1 So amodp r By properties of modulo operation V xiyifxmodp y modp henxnmudp ynmodp.fr all nEET So it suffices to prove that pPmed p r consider the group VCR L1,2 n p 19 where the group operation is multiplication modulo p By Cor 4 getmod p I By propertiesof modulo operation V a.bi.DE ifamodp bmodpand cmodp dmodp then acmadp abdmodp So rMmodp I made rmodp rmodp rPmodp rmodp h EB \documentclass[11pt]{article} \usepackage{verbatim} \usepackage[pdftex]{hyperref} \usepackage{amssymb} \usepackage{amsmath} \usepackage{graphicx} \voffset= -0.5in \textheight=8.8in \textwidth=6.5in \oddsidemargin=0in \parindent=0pt \begin{document} \pagestyle{empty} \begin{center} {\large \bf \sc MATH 421/521 Section B Intro to Abstract Algebra HW3 --- Fall 2020} \ \vspace{0.5em} \end{center} All homework are required to be typed in LaTeX.

You can use the free online editor\ \url{ See \url{ for a brief introduction. \ HW3 is due Tuesday September 29, by 11:59pm. Please upload your solutions on canvas under Assignments by the due time. \medskip 1. List the left cosets of the subgroups in each of the following. Here $\langle a\rangle$ denotes the subgroup generated by the element $a$. \medskip (a) $\langle 8 \rangle$ in $(\mathbb{Z}_{24},+)$ \medskip (b) $\langle 3 \rangle$ in $U(8)$ \medskip (c)