# Fourth Isomorphism Theorem

The proof of the fourth Isomorphism theorem ( Lattice Isomorphism Theorem) is done only upto proving the bijection between the sets of subgroups. I'm not clear how other points follow from this bijection. Please help.

• They don't. You need to prove additional properties of the (standard) bijection: that it preserves inclusions, indexes, quotients, normality, etc. Mar 5, 2019 at 10:02
• So how do I go about proving this theorem in detail? There's no book I could find helping me with this theorem. Mar 5, 2019 at 10:05
• So that depends on the particular statement of the Lattice Theorem you are dealing with. For example if $A\subseteq B$ then obviously $A/N\subseteq B/N$. Simply because if $a\in A$ then $aN\in A/N$. Other statements are either simple or follow from other isomorphism theorems. Anyway I suggest trying to prove it yourself. You can always come back here if you're stuck. Mar 5, 2019 at 10:58
• This is a rather basic result, and most books have it with different names and, perhaps, splitting it in several parts. You could check "correspondence Theorem", for example. It is theorem 3.7 in Isaacs' "Algebra, a Graduate Course", or theorem 2.28 in Rotman's "An Itroduction to the Theory of Groups" Mar 5, 2019 at 11:04