Given a function \( f \) mapping from a set \( X \) to a set \( Y \):

Otherwise known as a “one-to-one” function.

Given a function \( f \) mapping from a set \( X \) to a set \( Y \):

Otherwise known as an “onto” function.

A bijective function is one that is both injective and surjective.

A homomorphism is a map \( \phi \) from a group \( G \) to a group \( H \), for elements \( g,~ h \in G \), \( \phi \) is a homomorphism if and only if:

An isomorphism \( \phi \) from a group \( G \) to a group \( H \) is a bijective homomorphism.

An endomorphism is a homomorphism \( \phi \) mapping from a group \( G \) to itself.

An automorphism is a bijective endomorphism; i.e. an isomorphism from a group \( G \) to itself.

Two elements \( a,~ b \) of a group \( G \) are conjugates if there exists another element \( g \) in \( G \) such that \( b = gag^{-1} \) (i.e.,  \( \mu (\mu (g,~ a),~ \iota (g)) \) ).