Homomorphisms of Quaternion Group

[rmjmu507]

Homework Statement

Let Q = {±1, ±i, ±j, ±k} be the quaternion group. Find all homomorphisms from Z₂ to Q and from Z₄ to Q. Are there any nontrivial homomorphisms from Z₃ to Q?

Then, find all subgroups of Q.

Homework Equations

The Attempt at a Solution

I don't even know where to begin, I have never seen a problem like this before. Please help!

 

[brmath]

First, do you know what a group is? Assuming you do, how many elements are in this group? What is the binary operation * (addition, multiplication, or what?) And do you know how to compute a*b where a and b are in Q?

If you don't know those things, you really have to sort them out.

Next, what is a homomorphism? It is a map (which you can think of as a function), call it f, from the "source" group, call it G, to the "target" group Q, which preserves the relationships within the group. For example, if you know that x*y = z in G, it must be that f(x)*f(y) = f(z) in Q. An immediate observation is that if I is the identity element in G then f(I) has to be the identity element in Q.

Next, do you know what $Z_2$, $Z_3$ and $Z_4$ are? If you are not sure, look through your class notes or your textbook -- it's standard stuff and you can find it anywhere.

Once you tell me that you have all these definitions straight, we can move forward. Or if you have questions about them, ask.

 

[rmjmu507]

I know what a group is. The quaternion group mentioned above contains 8 elements. No operation is specified

Z2, Z3, Z4 are the cyclic groups of order 2, 3, 4 respectively.

 

[brmath]

The quaternion group would be under multiplication. Before you can be sure you have a homomorphism, and when you get to the question about the subgroups, you would have to know how to multiply two elements together in the quaternion group. I promise it's online if you don't know.

Once you are sure of that start with $Z_2$. It has two elements {1,a} and the 1 has to map into 1 in Q. So what are the possibilities for the a? If we had no constraints on the map f(a) could be anything, but we are constrained by the fact that it is a homomorphism. Can you work out which elements of Q a could conceivably map into?

Once you've done that, see what you can do with $Z_4$. It's only a little more complicated. Before you tackle the $Z_3$ question, make sure you know what the "trivial" homomorphism is.

I hope you are not offended that I asked if you know what a group is. Sometimes people are stuck on a question because they just don't know the basic definitions. So no offense was meant.

 

[Dick]

I know what a group is. The quaternion group mentioned above contains 8 elements. No operation is specified

Z2, Z3, Z4 are the cyclic groups of order 2, 3, 4 respectively.

The binary operation in the quaternion group usually called multiplication. Look it up to get the group operation rules, some are obvious, some aren't. The obvious starting point is that the identity in one group must map to the identity in the other group under a homomorphism.