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\(\def\Z{\mathbb{Z}} \def\zn{\mathbb{Z}_n} \def\znc{\mathbb{Z}_n^\times} \def\R{\mathbb{R}} \def\Q{\mathbb{Q}} \def\C{\mathbb{C}} \def\N{\mathbb{N}} \def\M{\mathbb{M}} \def\G{\mathcal{G}} \def\0{\mathbf 0} \def\Gdot{\langle G, \cdot\,\rangle} \def\phibar{\overline{\phi}} \DeclareMathOperator{\lcm}{lcm} \DeclareMathOperator{\Ker}{Ker} \def\siml{\sim_L} \def\simr{\sim_R} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Chapter9The Isomorphism Theorems

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.