Section 8 Groups of Permutations

8.5 Theorem Permutation Group

[[Math/ Books/A First Course in Abstract Algebra.pdf#page=82&selection=222,0,239,44&color=note|p.77]]

Let A be a nonempty set, and let $S_A$ be the collection of all permutations of A. Then $S_A$ is a group under permutation multiplication.

8.7 Example

Example

The nth dihedral group is defined as $D_n$ or the symmetries of a regular n-gon. $S_3\simeq D_3$

8.14 Definition Image

[[Math/ Books/A First Course in Abstract Algebra.pdf#page=87&selection=7,0,64,0&color=note|p.82]]

Let f : A → B be a function and let H be a subset of A. The image of H under f is $\{f(h)|h\in H\}$ and is denoted by $f[H]$.

8.16 Cayley’s Theorem

[[Math/ Books/A First Course in Abstract Algebra.pdf#page=87&selection=473,0,473,53&color=note|p.82]]

Every group is isomorphic to a group of permutations.

Or Every group is isomorphic a subgroup of a symmetric group $S_n$

THIS IS CRAZY

• Proof of Cayley’s Theorem 🔼