- #1
robertjordan
- 71
- 0
The definition given is...
"Let ##\phi: G \rightarrow H## be a homomorphism with kernel ##K##. The quotient group ##G/K## is the group whose elements are the fibers (sets of elements projecting to single elements of H) with group operation defined above: namely if ##X## is the fiber above ##a## and ##Y## is the fiber above b then the product of ##X## and ##Y## is defined to be the fiber above the product ##ab##."
But what if we have a homomorphism ##\alpha: G \rightarrow A## and a homomorphism ##\beta: G \rightarrow B## that both have kernel ##K##?
Wouldn't this mean ##G/K## is not unique because ##\alpha: G \rightarrow A## requires ##G/K## to be fibers consisting of elements in ##A## whereas ##\beta: G \rightarrow B## requires ##G/K## to be fibers consiting of elements in ##B##?
I'm confused. :(
"Let ##\phi: G \rightarrow H## be a homomorphism with kernel ##K##. The quotient group ##G/K## is the group whose elements are the fibers (sets of elements projecting to single elements of H) with group operation defined above: namely if ##X## is the fiber above ##a## and ##Y## is the fiber above b then the product of ##X## and ##Y## is defined to be the fiber above the product ##ab##."
But what if we have a homomorphism ##\alpha: G \rightarrow A## and a homomorphism ##\beta: G \rightarrow B## that both have kernel ##K##?
Wouldn't this mean ##G/K## is not unique because ##\alpha: G \rightarrow A## requires ##G/K## to be fibers consisting of elements in ##A## whereas ##\beta: G \rightarrow B## requires ##G/K## to be fibers consiting of elements in ##B##?
I'm confused. :(