List all isomorphisms (Z4,+) -> (Q,o), where Q is the set of all rotations of the square into itself with composition as binary operation. (Z4 is integers modulo 4 and o is the composition operation)
I'm just working ahead for my class, and I'm working as many problems as I can. This one confused me.
First off... I am really confused what this even looks like. What is Q exactly supposed to be really? And how does the composition function work with it? From the diagram I've drawn, this makes no sense to me, so I really wanted that cleared up. You don't really have to list all the isomorphisms for me to give you a top answer, but I do at least want to know what it's asking!
Just to say this so far, I haven't been introduced to groups yet. I say this because I asked another question on morphisms and people responded by talking about groups. So, I list the author's definition of a morphism here:
A morphism f:(X,*) -> (X',*') is defined to be a function on X to X' which "carries" the operation * on X into the operation *' on X', in the sense that f(x*y)=(fx)*'(fy) for all x,y in X.
I'm just working ahead for my class, and I'm working as many problems as I can. This one confused me.
First off... I am really confused what this even looks like. What is Q exactly supposed to be really? And how does the composition function work with it? From the diagram I've drawn, this makes no sense to me, so I really wanted that cleared up. You don't really have to list all the isomorphisms for me to give you a top answer, but I do at least want to know what it's asking!
Just to say this so far, I haven't been introduced to groups yet. I say this because I asked another question on morphisms and people responded by talking about groups. So, I list the author's definition of a morphism here:
A morphism f:(X,*) -> (X',*') is defined to be a function on X to X' which "carries" the operation * on X into the operation *' on X', in the sense that f(x*y)=(fx)*'(fy) for all x,y in X.
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Rotations of the square is just the cyclic group of order 4, generated by a 90 degree rotation. The elements are the 90 degree rotation, the 180 degree rotation, the 270 degree rotation and the identity. The group operation is composition, for example the 90 deg rotation combined with the 90 degree rotation mean rotate the square 90 degrees, twice, which is the same as the 180 degree rotation.
Seems to me you should be quite familiar with isomorphisms and homomorphisms before you worry about the more abstract notion of morphism.
Seems to me you should be quite familiar with isomorphisms and homomorphisms before you worry about the more abstract notion of morphism.