Kernel (Group)

up:: Group Homomorphism

Given a mapping between groups $\phi :G\to G^{\prime }$, its kernel is defined as its “null space”, i.e. all elements which are mapped to the identity

$$\ker \phi =\{k\in G\mid \phi (k)=e_{G^{\prime }}\}$$

Equivalently, it is the inverse image of the identity

$$\ker \phi ={\phi }^{-1}(e_{G^{\prime }})\subset G$$