Orders of elements in a quotient group.

[Artusartos]

Homework Statement

I want to find the orders of the elements in Z_8/(Z_4 \times Z_4), (Z_4 \times Z_2)/(Z_2 \times Z_2), and D_8/(Z_2 \times Z_2).

Homework Equations

The Attempt at a Solution

The elements of Z_2 \times Z_2 are (0,0), (1,0), (0,1), (1,1), and the elements of Z_8 are of course 1, 2, 3, 4, 5, 6, and 7.

So elements of Z_8/(Z_2 \times Z_2) must be of the form…

(0,0), (1,0), (0,1), (1,1)
1+(0,0), 1+ (1,0), 1+ (0,1), 1+ (1,1)
2+(0,0), 2+(1,0),2+ (0,1), 2+(1,1)
3+(0,0), 3+(1,0), 3+(0,1), 3+(1,1)
…
7+(0,0), 7+(1,0), 7+(0,1), 7+(1,1)

But since the number we are adding to Z_2 \times Z_2 is mod 8, none of these are equal. So I don’t know how we have 2 elements Z_2 \times Z_2.

I’m having the same problem with D_8/(Z_2 \times Z_2).<br /> We know that D_8 = \{e, b, b^2, b^3, a, ba, b^2a, b^3a \}<br /> <br /> (0,0), (1,0), (0,1), (1,1)<br /> b+(0,0), b+(1,0), b+ (0,1), b+(1,1)<br /> …<br /> b^3a+(0,0), b^3a+(1,0), b^3a+(0,1), b^3a+(1,1)<br /> <br /> But, for example, we know that b has order 4, right?

 

[Dick]

Homework Statement

I want to find the orders of the elements in Z_8/(Z_4 \times Z_4), (Z_4 \times Z_2)/(Z_2 \times Z_2), and D_8/(Z_2 \times Z_2).

Homework Equations

The Attempt at a Solution

The elements of Z_2 \times Z_2 are (0,0), (1,0), (0,1), (1,1), and the elements of Z_8 are of course 1, 2, 3, 4, 5, 6, and 7.

So elements of Z_8/(Z_2 \times Z_2) must be of the form…

(0,0), (1,0), (0,1), (1,1)
1+(0,0), 1+ (1,0), 1+ (0,1), 1+ (1,1)
2+(0,0), 2+(1,0),2+ (0,1), 2+(1,1)
3+(0,0), 3+(1,0), 3+(0,1), 3+(1,1)
…
7+(0,0), 7+(1,0), 7+(0,1), 7+(1,1)

But since the number we are adding to Z_2 \times Z_2 is mod 8, none of these are equal. So I don’t know how we have 2 elements Z_2 \times Z_2.

I’m having the same problem with D_8/(Z_2 \times Z_2).<br /> We know that D_8 = \{e, b, b^2, b^3, a, ba, b^2a, b^3a \}<br /> <br /> (0,0), (1,0), (0,1), (1,1)<br /> b+(0,0), b+(1,0), b+ (0,1), b+(1,1)<br /> …<br /> b^3a+(0,0), b^3a+(1,0), b^3a+(0,1), b^3a+(1,1)<br /> <br /> But, for example, we know that b has order 4, right?

<br /> <br /> Your first question doesn't make sense to me whatsoever. The quotient G/H is defined when H is a subgroup of G. Z_2 \times Z_2 is not a subgroup of $Z_8$. It doesn't even have a subgroup isomorphic to Z_2 \times Z_2. $D_8$ does so I think you can make some sense of that.

 

[Artusartos]

Your first question doesn't make sense to me whatsoever. The quotient G/H is defined when H is a subgroup of G. Z_2 \times Z_2 is not a subgroup of $Z_8$. It doesn't even have a subgroup isomorphic to Z_2 \times Z_2. $D_8$ does so I think you can make some sense of that.

Thanks a lot.