What Are the Cosets in Q/Z(Q)?

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Homework Help Overview

The problem involves finding the cosets in the context of the group Q/Z(Q), where Q is initially assumed to represent the quaternion group. The discussion centers around understanding the center of Q and its implications for determining cosets.

Discussion Character

  • Conceptual clarification, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants explore the definition of Q and its center, questioning whether Q refers to the quaternion group or another structure. There are discussions about the nature of the operation involved and the implications for the center Z(Q).

Discussion Status

Some participants have provided insights into the structure of Z(Q) and its elements, while others are clarifying assumptions about the group Q. There is a mix of interpretations regarding the nature of the operation and the elements involved, indicating an ongoing exploration of the topic.

Contextual Notes

There is uncertainty regarding the definition of Q and whether it refers to the quaternion group or the rational numbers. This ambiguity affects the understanding of Z(Q) and the subsequent determination of cosets.

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Homework Statement


Find the cosets in Q/Z(Q)


Homework Equations





The Attempt at a Solution



So Z(Q) is the centre of Q..
Then Z(Q) is normal in Q.

I don't get what the cosets would be without any given elements of Q or Z(Q)..
But I'm assuming since it is the centre of Q there is some trick?

Thanks.
 
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What is Q here? The quaternion group? I will assume it is.

What is the center of Q? Does i commute with every element? does j? does k? does -1? etc.? Once you have determined the center you should be able to find representatives for every coset in Q/Z(Q).
 
Hmm, is the operation simply addition? If so, Q is commutative, and Z(Q)=Q.
 
I get:
{1,-1} = Z(Q)= Z(Q)(-1)
{i, -i} = Z(Q)i = Z(Q)(-i)
{j, -j} = Z(Q)j = Z(Q)(-j)
{k, -k} = Z(Q)k = Z(Q)(-k)

is this correct?
 
Yes, this is correct. Sorry for the first answer, I thought Q ment rationals...
 

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