Endomorphism monoid

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Definition

The endomorphism monoid of a group is the monoid defined as follows:

  • Its elements are the endomorphisms of the group.
  • Its identity element is the identity map from the group to itself.
  • Multiplication is by function composition.

The elements of the endomorphism monoid with two-sided inverses form a subgroup of the endomorphism monoid and this subgroup is the automorphism group of the original group.