(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [itex]G[/itex] be a group with a normal subgroup [itex]N[/itex] and subgroups [itex] K \triangleleft H \leq G. [/itex]

If [itex]H/K [/itex] is nontrivial, prove that at least one of [itex]HN/KN[/itex] and [itex](H\cap N)/(K\cap N)[/itex] must be nontrivial.

2. Relevant equations

The Three (or Four) Isomorphism Theorems.

3. The attempt at a solution

By the first isomorphism theorem, we saw that [itex]HN/KN \cong H/K [/itex]. So if [itex]H/K[/itex] is nontrivial, then [itex]HN/KN [/itex] is also nontrivial.

Now to show that [itex](H\cap N)/(K\cap N)[/itex] is also nontrivial, what normal subgroup of [itex]H/K [/itex] is this quotient group [itex](H\cap N)/(K\cap N)[/itex] isomorphic to?

Because of the "and" in the statement of the problem, should both be nontrivial?

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# Homework Help: Some basic algebra (using Isomorphism Theorems)

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