Understanding butterfly lemma and Isomorphism theorems This is the lemma in the title: https://en.wikipedia.org/wiki/Zassenhaus_lemma
I've been studying isomorphism theorems and I have an intuition on them, and feel that they are quite natural. The motivation for these first theorems is clear, but not for butterfly one. What intuition could ge get for this lemma?  Any insight is appreciated.
 A: Have you checked this discussion? I am attaching/quoting the main parts below (with credits). If this is not allowed, please let me know or make the necessary changes (without informing me).

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*Following is (Serge) Lang's proof (from that page):



*Following is user Tab1e's proof from the same page:

This answer is based on https://math.berkeley.edu/~gbergman/.C.to.L/
asserting some of my understandings
If $G$ is a group, and $U/u$, $V/v$ are homomorphic images of
subgroups of $G$, meaning that there are two surjective
homomorphisms(Results of Fundamental theorem on homomorphisms):
one is from $U$ to $U/u$, and the other one is from $V$ to $V/v$.
After the definitions of the objects, one would like to describe the
extent to which one can "relate" part of the structure of $U/u$ and
part of the structure of $V/v$, based on their common origin in $G$.
To find the common "region", one can take the subgroup of $U/u$
consisting of all elements which are also images of elements of $V$,
which is written this as $u(U \cap V)/u$. Similarly, the analogous
subgroup of $V/v$ is $(U \cap V)v/v$.
To get a common homomorphic image of these, we must divide each by the
subgroup of those elements that are annihilated in the construction of
the other. Namely, use $u(U \cap v)$ instead of $u$ in the denumerator
and $(u \cap V)v$ instead of $v$ in the respective denumerator.
Now, the Butterfly Lemma says that after making these adjustments, we
do get isomorphic groups, the "common heritage" of $U/u$ and $V/v$.
Remark: Group-theorists call a factor-group of a subgroup of a group $G$ a subfactor of $G$. Thus, given two subfactors $U/u$ and
$V/v$ of a group $G$, the Butterfly Lemma characterizes their largest
natural "common subfactor".


P.S. Please do not gift me the bounty even if this answer is helpful.
