TITLE: Introducing a mathematical structure.
QUESTION [5 upvotes]: Most of the times, introducing a new mathematical structure is done in the following path.

Start with a set/collection, name it as $X$. It is possible that $X$ already have a structure with it, namely the structure of topological  space/manifold/vector space etc. 
Define a structure on $X$, denote it by $\mathcal{A}$. So, a structure is a pair $(X,\mathcal{A})$. 
Define what does it mean to say a substructure of $(X,\mathcal{A})$. Giving names to well-behaved substructures.
Define what are maps between two structures, say $(X,\mathcal{A})$ and $(Y,\mathcal{B})$. Giving names to well-behaved maps between two structures.
Define what does it mean to say two structures $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ are "equivalent".
Construct new structures from old structures. 

This step 6 differs drastically from one structure to another structure. Some ways to produce new structures from old structures are 

Quotients.
Pullbacks.
Products (direct).
Sums (direct).
Limits (Injective/Projective).
...
...

For example, defining the notion of structure of a group on a set, we follow the same procedure. Defining a group, defining what does it mean to say a group morphism, what it means to say a subgroup (a normal subgroup), quotients of subgroup, (direct) sum, (direct) product of groups, (Injective/Projective) limit of (a collection) of groups and so on.
In this post, I want to collect this procedure for most of the structures introduced in undergraduate or beginning graduate courses in Mathematics.

REPLY [2 votes]: Definition. Let $G$ be a set and let $(\cdot):G \times G \rightarrow G$ be a binary operation, called multiplication. Then $(G, \cdot)$ is a group if

Asocciativity: $(g_1g_2)g_3 = g_1(g_2g_3)$ for all $g_1, g_2, g_3 \in G$.
Identity: There exists an element $e$ such that $eg = ge = g$ for all $g \in G$.
Inverses: For any $g \in G$ there exists an element $g^{-1} \in G$ such that $gg^{-1} = g^{-1}g = e$.

Substructures. A subgroup of a group $G$ is a subset of $G$ that is itself a group and the inclusion is a group homomorphism (see below).
Maps. A group homomorphism is a map $f: G \rightarrow H$ between groups such that for all $g, h \in G$ we have $f(gh) = f(g)f(h)$. A group isomorphism is a bijective group homomorphism.
Constructions. The product group of a family $\{G_i\}_{i \in I}$ of groups is obtained as follows. The underlying set is the product $\prod_{i \in I} G_i$ and the group operation is coordinatewise multiplication.
The direct sum of a family $\{G_i\}_{i \in I}$ of groups is very similar in construction to the product, but differs in one important detail: The underlying set is the subgroup $\bigoplus_{i \in I}G_i$ of $\prod_{i \in I}G_i$ such that all but finitely many components are equal to the identity.
A normal subgroup of a group is a subset $N$ of $G$ such that $gNg^{-1} = N$ for all $g \in G$. In this case, the quotient group $G/N = \{gN : g \in G\}$ has multiplication $gN \cdot hN = ghN$. The requirement that $N$ is normal is equivalent to the requirement that the multiplication on $G/N$ is well-defined.