Dedekind's Modular Law states :
Let $G$ be a group and $H, K$ and $L$ be subgroups of $G$ with $K ≤ L$. Then $$\;HK ∩ L = (H ∩ L)K.$$
What is the reason that it is called modular law? Also, what is the intuition behind this theorem?
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This is two separate questions. – Shaun Dec 21 '20 at 16:30
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4It is a property of meet & join in the ordered structure (=lattice) of submodules of a module over any ring $R$. See Modular Lattice. It crops up, when studying various properties of the submodule lattice. Not unlike the better known parallelogram rule (aka 2nd Isomorphism theorem). Observe that it is a bit problematic when viewed as a statement about groups. For the product $HK$ is not always a subgroup. – Jyrki Lahtonen Dec 21 '20 at 16:38
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2In some sense the importance is that this prohibits sublattices isomorphic to the notorius $N_5$. In those lattice diagrams every line corresponds to a simple quotient. Forbidding $N_5$ then is a manifestation of Jordan-Hölder, or the module theoretic analogue of uniqueness of factorization. – Jyrki Lahtonen Dec 21 '20 at 16:43
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2I'm not sure those two comments constitute an answer to your question. Needs fleshing out anyway. May be later? Anyway, we probably have more knowledgable users who have used this more recently than I, so ... :-) – Jyrki Lahtonen Dec 21 '20 at 16:44
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Some intuition: abstract algebra - Why are modular lattices important? - Mathematics Stack Exchange – user202729 Mar 10 '23 at 16:32