Almost 20 years ago I was asked a question by Herbert Kociemba, a computer scientist who has one of the best Rubik’s cube solving programs known. Efficient methods of storing permutations in and (the groups of all permutations of the edges and vertices , respectively, of the Rubik’s cube) are needed, hence leading naturally to the concept of the complement of in . Specifically, he asked if has a complement in (this terminology is defined below). The answer is, as we shall see, ”no.” Nonetheless, it turns out to be possible to introduce a slightly more general notion of a ”-tuple of complementary subgroups” for which the answer to the analogous question is ”yes.”

This post is a very short summary of part of a paper I wrote (still unpublished) which can be downloaded here. This post explains the ”no” part of the answer. For the more technical ”yes” part of the answer, see the more complete pdf version.

**Notation**: If is any finite set then

- denotes the number of elements in .
- denotes the symmetric group on .
- denotes the symmetric group on .
- denotes its alternating subgroup of even permutations.
- denotes the cyclic subgroup of generated by the -cycle .
- denotes ”the Mathieu group of degree $10$” and order $720=10!/7!$, which we define as the subgroup of generated by and .
- denotes the Mathieu group of degree and order generated by and .
- denotes the Mathieu group of degree and order generated by and .
- For any prime power , denotes the finite field with elements.
- denotes the affine group of transformations on of the form , where and .

If is a finite group and are subgroups then we say is the **complement** of when

- , the identity of ,

and - .

Let denote a finite set. If is a subgroup of and then we let denote the **stabilizer** of in :

Let be a permutation group acting on a finite set (so is a subgroup of the symmetric

group of , ). Let be an integer and let

We say acts **-transitively on ** if acts transitively on via the ”diagonal” action

If acts transitively on and for some (hence all) then we say acts **regularly** on . If acts -transitively on and acts regularly on then we say acts **sharply -transitively** on .

The classification of -transitive groups, for , is to Jordan: A sharply -transitive group, , must be one of the following.

- : , and only.
- : , , and the Mathieu group .
- : , , and the Mathieu group .

We give a table which indicates, for small values of , which have a complement in .

complement of in | ||

size | ||

size | ||

size | ||

size | ||

size | ||

size | ||

size | ||

size | ||

( is another) | ||

size | ||

size | ||

size | ||

size | ||

**Proposition:** has a complement in if and only if there is an subgroup of such that

- acts -transitively on ,
- .

**Example:** has not one but two non-isomorphic subgroups, and , of order , each of which acts -transitively on . Thus has two non-isomorphic complements in .

The statement below is the main result.

**Theorem:** The following statements hold.

- If is not a prime power or a prime power plus then the only for which has a complement in are and .
- If is a prime power and not a prime power plus then the only for which has a complement in are , and .
- If is a prime power plus but not a prime power then the only for which has a complement in are , and .
- If is both a prime power plus and a prime power then the only for which has a complement in are , , and .
- If , see the above table.