Centerless group

Symbol-free definition
A group is said to be centerless if it satisfies the following equivalent conditions:


 * The center of the group is trivial.
 * The natural homomorphism from the group to its automorphism group that sends each element to the conjugation it induces, is injective (viz no two elements induce the same inner automorphism)

Definition with symbols
A group $$G$$ is said to be centerless if it satisfies the following equivalent conditions:


 * The center $$Z(G)$$ is the trivial group.
 * The natural homomorphism $$G \to \operatorname{Aut}(G)$$ given by $$g \mapsto c_g$$, where $$c_g = x \mapsto gxg^{-1}$$, is an injective map from $$G$$ to $$\operatorname{Aut}(G)$$.

Metaproperties
A direct product of centerless groups is centerless. This follows from the general fact that the center of a direct product equals the direct product of the individual centers.