Lctu 3.2: Costs Matthw Macauly Dpatmnt o Mathmatical Scincs Clmson Univsity http://www.math.clmson.du/~macaul/ Math 4120, Modn Algba M. Macauly (Clmson) Lctu 3.2: Costs Math 4120, Modn Algba 1 / 11
Ovviw Th gulaity popty o Cayly diagams implis that idntical copis o th agmnt o th diagam that cospond to a subgoup appa thoughout th st o th diagam. Fo xampl, th ollowing igus highlight th patd copis o = {, } in D 3: 2 2 2 2 2 2 Howv, only on o ths copis is actually a goup! Sinc th oth two copis do not contain th idntity, thy cannot b goups. Ky concpt Th lmnts that om ths patd copis o th subgoup agmnt in th Cayly diagam a calld costs. M. Macauly (Clmson) Lctu 3.2: Costs Math 4120, Modn Algba 2 / 11
An xampl: D 4 Lt s ind all o th costs o th subgoup H =, 2 = {,, 2, 2 } o D 4. I w us 2 as a gnato in th Cayly diagam o D 4, thn it will b asi to s th costs. Not that D 4 =, =,, 2. Th costs o H =, 2 a: H =, 2 = {,, 2, 2 }, }{{} H =, 2 = {, 3,, 3 }. }{{} oiginal copy 3 3 3 3 2 2 2 2 M. Macauly (Clmson) Lctu 3.2: Costs Math 4120, Modn Algba 3 / 11
Mo on costs Dinition I H is a subgoup o G, thn a (lt) cost is a st ah = {ah : h H}, wh a G is som ixd lmnt. Th distingusihd lmnt (in this cas, a) that w choos to us to nam th cost is calld th psntativ. Rmak In a Cayly diagam, th (lt) cost ah can b ound as ollows: stat om nod a and ollow all paths in H. Fo xampl, lt H = in D 3. Th cost {, } o H is th st H = = {, } = {, }. Altnativly, w could hav wittn ( )H to dnot th sam cost, bcaus 2 H = {, } = {, 2 } = {, }. 2 M. Macauly (Clmson) Lctu 3.2: Costs Math 4120, Modn Algba 4 / 11
Mo on costs Th ollowing sults should b visually cla om th Cayly diagams and th gulaity popty. Fomal algbaic poos that a not don h will b assignd as homwok. Poposition Fo any subgoup H G, th union o th (lt) costs o H is th whol goup G. Poo Th lmnt g G lis in th cost gh, bcaus g = g gh = {gh h H}. Poposition Each (lt) cost can hav multipl psntativs. Spciically, i b ah, thn ah = bh. Poposition All (lt) costs o H G hav th sam siz. M. Macauly (Clmson) Lctu 3.2: Costs Math 4120, Modn Algba 5 / 11
Mo on costs Poposition Fo any subgoup H G, th (lt) costs o H patition th goup G. Poo W know that th lmnt g G lis in a (lt) cost o H, namly gh. Uniqunss ollows bcaus i g kh, thn gh = kh. Subgoups also hav ight costs: Ha = {ha: h H}. Fo xampl, th ight costs o H = in D 3 a (call that = 2 ) and H = = {, } = {, } = {, 2 } 2 = {, } 2 = { 2, 2 } = { 2, }. In this xampl, th lt costs o a dint than th ight costs. Thus, thy must look dint in th Cayly diagam. M. Macauly (Clmson) Lctu 3.2: Costs Math 4120, Modn Algba 6 / 11
Lt vs. ight costs Th lt diagam blow shows th lt cost in D 3: th nods that aows can ach at th path to has bn ollowd. Th ight diagam shows th ight cost in D 3: th nods that aows can ach om th lmnts in. 2 2 2 2 Thus, lt costs look lik copis o th subgoup, whil th lmnts o ight costs a usually scattd, bcaus w adoptd th convntion that aows in a Cayly diagam psnt ight multiplication. Ky point Lt and ight costs a gnally dint. M. Macauly (Clmson) Lctu 3.2: Costs Math 4120, Modn Algba 7 / 11
Lt vs. ight costs Fo any subgoup H G, w can think o G as th union o non-ovlapping and qual siz copis o any subgoup, namly that subgoup s lt costs. Though th ight costs also patition G, th cosponding patitions could b dint! H a a w visualizations o this ida: g nh g n 1H H... g 1H g 2H g nh. g 2H g 1H Hg 2 Hg n... Hg 1 H H Dinition I H < G, thn th indx o H in G, wittn [G : H], is th numb o distinct lt (o quivalntly, ight) costs o H in G. M. Macauly (Clmson) Lctu 3.2: Costs Math 4120, Modn Algba 8 / 11
Lt vs. ight costs Th lt and ight costs o th subgoup H = D 3 a dint: 2 H H 2 2 H 2 2 H 2 H H Th lt and ight costs o th subgoup N = D 3 a th sam: N 2 N 2 N 2 N 2 Poposition I H G has indx [G : H] = 2, thn th lt and ight costs o H a th sam. M. Macauly (Clmson) Lctu 3.2: Costs Math 4120, Modn Algba 9 / 11
Costs o ablian goups Rcall that in som ablian goups, w us th symbol + o th binay opation. In this cas, lt costs hav th om a + H (instad o ah). Fo xampl, lt G = (Z, +), and consid th subgoup H = 4Z = {4k k Z} consisting o multipls o 4. Th lt costs o H a H = {..., 12, 8, 4, 0, 4, 8, 12,... } 1 + H = {..., 11, 7, 3, 1, 5, 9, 13,... } 2 + H = {..., 10, 6, 2, 2, 6, 10, 14,... } 3 + H = {..., 9, 5, 1, 3, 7, 11, 15,... }. Notic that ths a th sam th th ight costs o H: H, H + 1, H + 2, H + 3. Do you s why th lt and ight costs o an ablian goup will always b th sam? Also, not why it would b incoct to wit 3H o th cost 3 + H. In act, 3H would pobably b intptd to b th subgoup 12Z. M. Macauly (Clmson) Lctu 3.2: Costs Math 4120, Modn Algba 10 / 11
A thom o Josph Lagang W will inish with on o ou ist majo thoms, namd at th poliic 18th cntuy Italian/Fnch mathmatician Josph Lagang. Lagang s Thom Assum G is init. I H < G, thn H divids G. Poo Suppos th a n lt costs o th subgoup H. Sinc thy a all th sam siz, and thy patition G, w must hav G = H + + H = n H. }{{} n copis Tho, H divids G. Coollay I G < and H G, thn [G : H] = G H. M. Macauly (Clmson) Lctu 3.2: Costs Math 4120, Modn Algba 11 / 11