You just map the value (1, 1)

C_{m} \times C_{n}

to the value 1 of

C_{m n}

, and it works out. E.g. for

C_{2} \times C_{3}

, the group generated by of (1, 1) is:

0 = (0, 0)
1 = (1, 1)
2 = (0, 2)
3 = (1, 0)
4 = (0, 1)
5 = (1, 2)
6 = (0, 0) = 0

Cycle notation

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_{1} 1

is generated by three elements:

(0123456789a)
(0b)(1a)(25)(37)(48)(69)
(26a7)(3945)

which feels quite compact for a simple group with 95040 elements, doesn't it!

Stabilizer (group)

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:

Symmetric group

Group of all permutations.

All groups are isomorphic to a subgroup of the symmetric group

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

Alternating group ( $A_{n}$ )

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.