Finally, a valid generalisation of the Third Isomorphism Theorem

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Let G be a commutative group. Let M and N be subgroups of G. If M\leq N, then \frac{(G/M)}{(N/M)}\cong \frac{G}{N}.

However, what if M\not\leq N? We will consider the general scenario, where M and N are any subgroups of G, provided N is not a proper subgroup of M.

Then \frac{(G/M)}{(N/M)}\cong \frac{G}{N+M}

In the case that M\leq N, we have M+N=N.

I have arrived upon this result myself. I don’t know if it is already known.

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Graduate student

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