Multiplication table help Algebra

[elle]

Multiplication table help please! Algebra ...

Hi, I've come across this question in my algebra txtbook and the exercise asks to construct a multiplication table, verifying that the Quaternion Group forms a group under matrix multiplication.

http://i12.tinypic.com/2i20wus.jpg

I'm confused on how this works so let's say if I took I x (multiplied by) I that would just give the original matrix of I right? And if I x J then it would just be J ?

 

[nocturnal]

I'm confused on how this works so let's say if I took I x (multiplied by) I that would just give the original matrix of I right? And if I x J then it would just be J ?

Yes, exactly. Now calculate the rest of all the possible products.

 

[elle]

Okay I've filled in the table nows but all I need to do is verify with a statement that it forms a group under matrix multiplication. Can anyone help me with the statement? Do I just say its because it is closed since the elements all lie in the Quaternion group?

Thanks in advance

 

[nocturnal]

You need to show that the binary structure fits the definition of a group.
ie: show

1) multiplication is associative
2) existence of identity
3) existence of inverses

 

[elle]

uh...how do you show that? Could you give me an example say for 1) closure under x?

 

[nocturnal]

This is a finite group of 8 elements and the multiplication table should give all possible products. So for example to show each element has inverses, simply list the inverse for each elelment. You can refer to your multiplication table to verify this.

You don't really need to show anything for closure other than to state that for any two elements, their product is again an element, which should be obvious from the table.

 

[elle]

Oh okay I get it nows, thanks very much!