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The existing answers are about groups, but here is a more general perspective.

Suppose we have some kind of structure $S$ with a substructure $T$, meaning that $T$ inherits the operations and predicates from $S$ and is closed under those operations. We could ask the general question of whether we can partition $S$ into parts according to some equivalence relation $\sim$ such that $T$ is one of the parts, and all the operations and predicates commute with $\sim$. We can think of the partition as a projection from elements to parts.

For example, for a binary operation $\circ$ on $S$, we want to have $x \circ y \sim x' \circ y'$ if $x \sim x'$ and $y \sim y'$. And for a predicate $P$ on $S$, we want to have $P(x) \equiv P(x')$ if $x \sim x'$.

First let us see what we get if we do have such a partition.

The commuting of the operations and predicates with the partition implies that we can define the same operations and predicates on the parts themselves, simply by going forward and backward along the projection. Symbolically, for every $A,B \in S/\sim$ we define $P(A) \overset{def}\equiv ( P(x) \text{ where } x \in A )$ and $A \circ B \overset{def}= ( ( x \circ y ) / \sim \text{ where } x \in A \land y \in B )$. These definitions are valid precisely because the choice of representatives does not matter. (Here $x/\sim$ is naturally defined as the part containing $x$.)

If $\circ$ is associative with identity $e$ and inverse function $i$, then each part $A$ is generated by the action of $T$ on any one member. Symbolically, for any $a \in A$ we have $A = a \circ T \overset{def}= \{ a \circ x : x \in T \}$ and by symmetry $A = T \circ a$. This is one crucial point. Let us see why this is true. Note that $e \in T$ because $T$ is a substructure. For any $x \in T$ it is clear that $a \circ x \sim a \circ e = a$ and hence $a \circ x \in A$. Also, for any $a' \in A$ we have $i(a) \circ a' \sim i(a) \circ a = e \in T$, and hence $i(a) \circ a' = y$ for some $y \in T$, giving $a' = a \circ y$.

If we have just one such operation $\circ$, we get the isomorphism theorems in terms of cosets. For the first isomorphism theorem, any structure homomorphism $φ$ on $S$ would induce such a partition via ( $x \sim x'$ iff $φ(x) = φ(x')$ ), where $T = Ker(φ) \overset{def}= \{ x : x \in S \land φ(x) = φ(e) \}$, and hence the typical definition of the quotient structure $S / T$ as the cosets $\{ a \circ T : a \in S \}$ with the appropriate structure inherited from $S$ makes total sense since it is just $S / \sim$, which is obviously isomorphic to $Im(φ)$. Similar kind of reasoning gives the other isomorphism theorems.

Next let us see what we need to get such a partition.

All the following structures have an associative invertible operation $\circ$, and hence the crucial point above applies to all the following structures. Thus we know that the desired partition must be given by $x \sim y \overset{def}\equiv x \circ T = y \circ T$. Therefore, if $x \circ T = T \circ x$ for every $x \in S$, then given any $x \sim x'$ and $y \sim y'$ we have $x \circ y \circ T = x \circ y \circ T \circ T = x \circ T \circ y \circ T$ $= x' \circ T \circ y' \circ T = \cdots = x' \circ y' \circ T$, and hence $x \circ y \sim x' \circ y'$. So the algebraic identity ( $x \circ T = T \circ x$ ) is equivalent to the commuting of $\circ$ with the desired $\sim$. This is the second crucial point.

For a group $(G,*)$ to be partitioned by subgroup $(H,*)$, it is necessary and sufficient to have $g*H = H*g$ for every $g \in G$, by the second crucial point. And equivalently, $g*H*g^{-1} \subseteq H$ for every $g \in G$. This is precisely why the definition of normal subgroups works!

For a ring $(R,+,*)$ to be partitioned by subring $(S,+,*)$, we need to have $r+S = S+r$ for every $r \in R$, but that is automatic since $R$ is a ring. We also need to have for every $x,y \in R$ that $(x+S)*(y+S) = x*y+S$, equivalently $x*y+x*S+S*y+S = x*y+S$. This would be satisfied if we have $x*S = y*S = S$. This is the reason for the definition of ideals !

For an $R$-module $(M,+)$ to be partitioned by submodule $(M,+)$, it is completely automatic by commutativity of addition.

The notion of structure-preserving partitions is a common one. Vector spaces provide a convenient visual example, since such partitions of them are literally projections onto vector subspaces. The first isomorphism theorem here is equivalent to the rank-nullity theorem.

Another important example is the real numbers. They can be constructed from the structure $S$ of Cauchy sequences of rationals with pointwise addition and multiplication, by partitioning them according to whether their difference tends to zero. The pointwise operations clearly commute with the partition, and hence the ring properties carry straight over from $S$ to the reals. We can also define $<$ on $S$ via ( $f < g$ iff $f(n) < g(n)$ as $n \to \infty$ ), which we can prove is well-defined and a total order, and that it commutes with the partition. Thus it also carries over to the reals. The only really special fact to be proven is the existence of multiplicative inverse for a nonzero real. Well, the zero real is the part of $S$ containing the zero sequence, and every nonzero real is a part of $S$ whose sequences are eventually bounded away from zero. So the partitioning has effectively collapsed all the 'non-invertible' sequences in $S$ to a single zero real.

There are also many results in abstract algebra where normal subgroups are incredibly important. For example, for any finite normal field extension $K / F$ and any subgroup $G$ of $Aut(K/F)$ (the automorphisms of $K$ that fixes elements of $F$), we have that $G$ is a normal subgroup of $Aut(K/F)$ iff the elements of $K$ fixed by $G$ is a normal extension of $F$. And the unsolvability of the quintic turns out to be because the alternating group $A_5$ has no non-trivial normal subgroup.