Important mathematical group

You just map the value (1, 1) $C_m\times C_n$ to the value 1 of $C_{mn}$, and it works out. E.g. for $C_2\times C_3$, the group generated by of (1, 1) is:

A concise to describe a specific permutation.

A permutation group can then be described in terms of the generating set of a group of specific elements given in cycle notation.

E.g. en.wikipedia.org/w/index.php?title=Mathieu_group&oldid=1034060469#Permutation_groups mentions that the Mathieu group $M_{11}$ is generated by three elements:which feels quite compact for a simple group with 95040 elements, doesn't it!

Suppose we have a given permutation group that acts on a set of n elements.

If we pick k elements of the set, the stabilizer subgroup of those k elements is a subgroup of the given permutation group that keeps those elements unchanged.

Note that an analogous definition can be given for non-finite groups. Also note that the case for all finite groups is covered by the permutation definition since all groups are isomorphic to a subgroup of the symmetric group

TODO existence and uniqueness. Existence is obvious for the identity permutation, but proper subgroup likely does not exist in general.

Bibliography:

Group of all permutations.

Or in other words: symmetric groups are boring, because they are basically everything already!

Group of even permutations.

Note that odd permutations don't form a subgroup of the symmetric group like the even permutations do, because the composition of two odd permutations is an even permutation.

www.youtube.com/watch?v=U_618kB6P1Q GT18.2. A_n is Simple (n ge 5) by MathDoctorBob (2012)

Our notation: $D_n$, called "dihedral group of degree n", means the dihedral group of the regular polygon with $n$ sides, and therefore has order $2n$ (all rotations + flips), called the "dihedral group of order 2n".

17 of them.

All possible repetitive crystal structures!

219 of them.