What are some nice informations about the dihedral groups,alternating groups,symmetric groups.

Question

I am an undergraduate student and I want to know some nice informations about some special groups like the dihedral group $D_{2n}$,of regular $n$-gon , alternating group $A_n$ and symmetric group $S_n$.For example I am looking for some important informations like centre,normal subgroups,order of elements in it,normal subgroups.

These properties clearly say something about the structure of these groups.It is often helpful in solving problems of homomorphism and isomorphisms.

Can someone please provide me some necessary information I should know about these groups to tackle challenging problems on group theory?

Answer 1

Just a few links from the wealth of MSE posts:

1.) Center:

Find the center of the symmetry group $S_n$.

The center of $A_n$ is trivial for $n \geq 4$

Center of dihedral group

2.) Normal subgroups

Normal subgroups of the symmetric group $S_N$

Normal subgroups of the Alternating group $A_n$

Normal subgroups of dihedral groups

3.) Solvability and Nilpotency

Is the dihedral group $D_n$ nilpotent? solvable?

Answer 2

Every element of $S_n$ can be written uniquely as a product of disjoint cycles.

Conjugation preserves cycle structure.

The center of $S_n$ is trivial for $n\ge3$.

Every group can be embedded in a symmetric group (Cayley's theorem).

For $n\gt4$, $A_n$ is simple.

The smallest non-abelian group is $S_3$. And $S_3\cong D_3$.

$A_4$ has no subgroup of order $6$. So the converse to Lagrange fails.

$D_n$ is centerless for $n$ odd. The center of $D_{2n}$ is $\{e,r^n\}$.

These are just a few for starters. There are lots.