# Group Theory/Groups with structure

**Definition (group with structure)**:

Let be a concrete category. Then the category of **groups with structure** is the subcategory of which is defined as follows:

- Its objects are the objects of such that the categorical product exists (and, on a set level, equals the set-theoretic product of with itself, projections included) and whose underlying sets bear a group structure such that the group law is a morphism in and inversion is a morphism in .
- Its morphisms are morphisms in that, on the set level, are also group homomorphisms.

**Proposition (multipication by an element is an automorphism in the underlying concrete category for every group with structure in a category admitting enough constant morphisms)**:

Let be a group with structure whose additional structure is given by the concrete category , such that every constant morphism is a morphism of . Further, suppose that . Then the function

is an automorphism of in the category .

**Proof:** If we show that the given function is a morphism, we've completed the proof, since an inverse is given by left multiplication by .

Thus, consider the morphism whose first component is given by the constant function associated to and whose second component is given by the identity on : Postcomposing it with the group law yields the morphism in the theorem statement, which is hence a morphism of .