\sectionSection 4.1: Group Actions and Permutation Representations
Documenting your Dummit and Foote Chapter 4 solutions on Overleaf is a rigorous way to master Group Theory. It forces you to understand the logic behind every Sylow -subgroup and group action. dummit+and+foote+solutions+chapter+4+overleaf+full
\subsection*Exercise 15 Prove that there is no simple group of order $56 = 2^3\cdot 7$. \sectionSection 4
If you just need to check your work, several sites host pre-compiled PDFs of Chapter 4 exercises: Greg Kikola's Website If you just need to check your work,
Use \counterwithinexercisesection to get labels like "Exercise 4.2.7".
\beginenumerate[label=(\roman*)] \item For any prime $p$ dividing $|G|$, $G$ has a Sylow $p$-subgroup (of order $p^a$ where $p^a \mid |G|$ but $p^a+1\nmid |G|$). \item All Sylow $p$-subgroups are conjugate. The number $n_p$ of Sylow $p$-subgroups satisfies $n_p \equiv 1 \pmodp$ and $n_p \mid |G|/p^a$. \item Any $p$-subgroup of $G$ is contained in some Sylow $p$-subgroup. \endenumerate