Use the Fundamental Theorem of Homomorphism for Groups to prove

Question

Use the Fundamental Theorem of Homomorphism for Groups to prove the following theorem, which is called the Zassenhaus (Butterfly) Lemma:
Let H and K be subgroups of a group G and H' and K' be normal subgroups of H and K, respectively. Then
i) $H'(H \cap K') \triangleleft H'(H \cap K)$
ii) $K'(H' \cap K) \triangleleft K'(H \cap K)$

iii) $\frac{H'(H \cap K)}{H'(H \cap K')} \simeq \frac{K'(H \cap K)}{K'(H' \cap K)} \simeq \frac{(H \cap K)}{(H' \cap K)(H \cap K')}$ $\qquad \qquad \qquad \qquad \qquad \qquad \quad$
The situation can be represented by the subgroup diagram below, which explains the name ‘butterfly’.

Image ignouassignments-ignouacademy-com--p-theorem-10733

PART-B (MM: 30 Marks)
(Based on Block 3.)

10 Jan 2026

Answer :

Word Count : 218

It is the subgroup lattice (Butterfly Diagram) used to illustrate the Zassenhaus (Butterfly) Lemma in Group Theory. ### Labels in the diagram * Top left: (H) * Top right: (K) * Middle top left: (H'(H\cap K)) _______ ____ ______ ______ __________ ________ ____ ________ ____ ______.
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