# Group Theory/Groups with structure

Definition (group with structure):

Let ${\displaystyle {\mathcal {C}}}$ be a concrete category. Then the category of groups with structure is the subcategory of ${\displaystyle {\mathcal {C}}}$ which is defined as follows:

• Its objects are the objects ${\displaystyle G}$ of ${\displaystyle {\mathcal {C}}}$ such that the categorical product ${\displaystyle G\times G}$ exists (and, on a set level, equals the set-theoretic product of ${\displaystyle G}$ with itself, projections included) and whose underlying sets bear a group structure such that the group law is a morphism ${\displaystyle G\times G\to G}$ in ${\displaystyle {\mathcal {C}}}$ and inversion is a morphism ${\displaystyle G\to G}$ in ${\displaystyle {\mathcal {C}}}$.
• Its morphisms are morphisms ${\displaystyle f:G\to G'}$ in ${\displaystyle {\mathcal {C}}}$ 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 ${\displaystyle G}$ be a group with structure whose additional structure is given by the concrete category ${\displaystyle {\mathcal {C}}}$, such that every constant morphism ${\displaystyle G\to G}$ is a morphism of ${\displaystyle {\mathcal {C}}}$. Further, suppose that ${\displaystyle g\in G}$. Then the function

${\displaystyle G\to G,h\mapsto gh}$

is an automorphism of ${\displaystyle G}$ in the category ${\displaystyle {\mathcal {C}}}$.

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 ${\displaystyle g^{-1}}$.

Thus, consider the morphism ${\displaystyle G\to G\times G}$ whose first component is given by the constant function associated to ${\displaystyle g}$ and whose second component is given by the identity on ${\displaystyle G}$: Postcomposing it with the group law yields the morphism in the theorem statement, which is hence a morphism of ${\displaystyle {\mathcal {C}}}$. ${\displaystyle \Box }$