Understanding butterfly lemma and Isomorphism theorems

Question

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.

Answer 1

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

2. 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".

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