Fiber (Group Theory)

fiber

This is the preimage $f^{- 1}$ of some singleton set ${y}$ . Namely:

f^{- 1} ({y}) = {x : f (x) = y}

Contains all the sets of inputs that give the singleton provided.

Note that for a specific Homomorphism (or Group Morphism) $φ$ then we have that the fiber above $a$ is:

φ^{- 1} (a) = {g \in G | φ (g) = a}

so it's all the $g$ 's in our Group that map to $a$ .

Further, sometimes mathematicians will drop the set notation instead which is technically wrong:

φ^{- 1} (a) \equiv φ^{- 1} ({a})

even though they are technically not the same thing.