Question

Let G be a cyclic group with generator a, |a|=20 and H=.Find the order of a^3H in G/H.

Answer 1

\[G=\left\{ e,a ^{1},a ^{2},a ^{3},a ^{4},a ^{5},... a ^{19}\right\}\]

Answer 2

i found that\[<a ^{4}>=\left\{ e,a ^{4},a^{8},a ^{12},a ^{16} \right\}\]
This means
\[a ^{3}<a ^{4}>=\left\{ a ^{3},a ^{7},a ^{11},a ^{15},a ^{19} \right\}=a ^{3}H\]

Answer 3

THEN WHAT WILL BE THE ORDER OF \[a ^{3}H\]

Answer 4

@terenzreignz

Answer 5

is any element of a^3 (except e) in the subgroup <a^4> ?

Answer 6

no

Answer 7

of course there is :)
a^12

Answer 8

not all of them are in <a^4>

Answer 9

just one is needed...
a^3 <a^4>
a^6 <a^4>
a^9 <a^4>
a^12 <a^4> = <a^4>
order is 4.

Answer 10

i thought they want the smallest integer such that a^n =e

Answer 11

that would be the order of a, but that's not what they're asking :)
They're asking the smallest integer such that
a^3n <a^4> = <a^4>

Answer 12

so wat would be the difference if they have said find the order ofa^3H

Answer 13

big difference. as was demonstrated, the order of a^3 <a^4> is 4

Answer 14

G/H={H,aH,a^2H,a^3H} so wat is the purpose of ths statement on this question

Answer 15

so it means even if they have said . Find the order of a^3H we will get the same answer

Answer 16

by excluding in G/H

Answer 17

we're just finding the the order of a^3H
considering that <a^3 H> is a subgroup of G/H

Answer 18

OK :)

Answer 19

DID U DO THE EQUIVALENCE RELATION