Question

Abstract Algebra

Find all left cosets of <up> in D4

Answer 1

For Du and according to question, Lus> b=uo and a=f So tef cosets of Luty are: {e, of}, {I, Juf}, {92, J² uf} and {f}, 13 of}

Given o dan G= Dihedral group dan Sea, a? ..., ahla, beG Ib, ab, a²b, ..., anbe when ocasan and o(b)=2 ire a dihedral group dan is always generated by two elements one rotation (a) and reflection 6). Here b, ab, a²b, - a., ahib be are in numbers of reflection and each has onder 2 So the subgroup of reflection is always of the form: H=&e, x} where ne {b, ab, a²b, ..., amb} Without loss of generality take usb. so Hage, by so add the left Eoset of H has only à elements each. So there will be total= 20 = n number of left cosets of H. The left cosets of H are of the formi a H= {a", aby where oLiLn So the list of all left cosets of it are' {e, b}, {a, ab} {ar, a²b2 - - a., {ann, ab? On the above there are int deft cosets and each are disjonit which can be proved below: bet hat, aby = {a', a b} a = a or a = alb. - aci= e or a = b contradiction as Contaddiction as o@=n) bisa reflection (not lie in rotation! so and he, of, sa, an}, {a} arb} - .. salah-16} are all distinct required in number of left cosets of reflection subgroup H.

Given G=Dehedral gToup dan Sa, at,a'- b, ab, a'b wheT oa) n and ob)= 2 dan is always generated by a dihedhra i-e two elemunts one rotatiOn (a) and refDection Here b, ab, a' b, arb be are in' number f -- veflection ond each has onder 2 Sa the subgroup of eflectian is aluys of the fem H-fe, n3were Re bab,ab, a7 Without Less of genera lity taxe x-b Sa So afl the lett Eoset ot H has ondy a elements euch. there wi IL be tatal- an n'number f lett of H. ase ts The lett cosets f H age of the form aiH-fa,ab ere oiin S. the list of all left cogets st H are' an-l an the abore there are n left cosats and each are disjonit which can be pveved belaw > ai-ie (comtnadiction as o@n) Fb Contradic toy as not 2ie cn otation Co ol fe,1,fa, a3,i a'i - are all dis tinct e1uire d n number 6lett Cosets t Teflection subaaoup H