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# Chapter9The Isomorphism Theorems¶ permalink

Recall that our goal here is to use a subgroup of a group $G$ to study not just the structure of the subgroup, but the structure of $G$ outside of that subgroup (the ultimate goal being to get a feeling for the structure of $G$ as a whole). We've further seen that if we choose $N$ to be a normal subgroup of $G\text{,}$ we can do this by studying both $N$ and the factor group $G/N\text{.}$ Now, we've noticed that in some cases—in particular, when $G$ is cyclic–it is not too hard to identify the structure of a factor group of $G\text{.}$ But what about when $G$ and $N$ are more complicated? For instance, we have seen that $SL(5,\R)$ is a normal subgroup of $GL(5,\R)\text{.}$ What is the structure of $GL(5,\R)/SL(5,\R)\text{?}$ That is not so easy to figure out by looking directly at left coset multiplication in the factor group.