Order of 5 + <3> in II18/<3> is 6

[aliciaislol]

Homework Statement

Find the order of the element 5 + <3> in the quotient group II₁₈/ <3>.

Homework Equations

II₁₈ = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}
<3> = {3,6,9,12,15,18}
5 + <3> = {5,8,11,14,17,2}
order is the number of elements

The Attempt at a Solution

I am thinking this II₁₈/ <3> equals {3,6,9,12,15,18} and I have to find the order of that plus 5.

So I got the order as 6.

 

[Billy Bob]

Find the order of the element 5 + <3> in the quotient group II₁₈/ <3>.

I assume II₁₈ means Z₁₈, i.e. integers mod 18 under addition mod 18.

II₁₈ = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}
<3> = {3,6,9,12,15,18}
5 + <3> = {5,8,11,14,17,2}

OK, and 18=0=the identity in II₁₈.

order is the number of elements

Not in this case. Order of a group is the number of elements. Order of an element is the smallest positive integer n such that a^n=aa...a=identity (in multiplicative notation), i.e. such that na=a+a+...+a=identity (in additive notation, which applies to this example). You have to find the order of the element 5 + <3> in the group II₁₈/<3>

I am thinking this II₁₈/ <3> equals {3,6,9,12,15,18} and I have to find the order of that plus 5.

So I got the order as 6.

This part is all wrong. II₁₈/ <3> equals {<3>,1+<3>,2+<3>}. The identity in this group is <3>, or maybe you call it 18+<3>.

Add the element 5+<3> to itself repeatedly until you get the identity <3>. Note that 5+<3> = 2+<3>. How many times did you add it to itself? That is, what was the smallest positive integer n such that n times 5+<3> equals the identity <3>? That n is your answer.

 

[aliciaislol]

so I did that and I came up with:
0+<3> = {0,3,6,9,12,15}
1+<3> = {1,4,7,10,13,16}
2+<3> = {2,5,8,11,14,17}
3+<3> = {3,6,9,12,15,0}
4+<3> = {4.7.10.13.16.1}
5+<3> = {5,8,11,14,17,2}
So I get that 0 & 3 are tha same, 1&4 are the same and 2&5 are the same.
So I have to do n(5+<3>)= {3,6,9,12,15,0}? b/c nothin times 2 will give me 15.