Math 360, Fall 2021, Assignment 12

"When I think of Euclid even now, I have to wipe my sweaty brow."

- C. M. Bellman

Carefully define the following terms, then give one example and one non-example of each:

1. Transposition (also known as a swap).

Carefully state the following theorems (you do not need to prove them):

1. Formula expressing any cycle as a product of transpositions.
2. Theorem concerning generation of $S_n$ by transpositions ("Any subgroup of $S_n$ which contains all of the transpositions is...").

Solve the following problems:

1. Express the cycle $(1,2,3,4,5)$ as a product of transpositions.
2. Express the cycle $(2,3,4,5,1)$ as a product of transpositions.
3. Observe that the cycles introduced in the previous two problems are in fact the same permutation. Is the decomposition of a permutation as a product of transpositions unique?
4. (Optional) Skim through the wikipedia article on the geometric concept of a simplex, paying particular attention to the section on the "standard" simplex, which is regular in the sense described in the article. (For fun, see also the article on the regular four-dimensional simplex, paying particular attention to the beautiful image of a rotating projection of this object into three-dimensional space.) Finally, show that the symmetry group of the standard $n$-simplex is isomorphic to $S_{n+1}$. (Hint: for any pair (i,j), the map that swaps the $i$th and $j$th coordinates is an isometry of $\mathbb{R}^n$.)
5. Referring to the result of the above exercise, calculate the symmetry groups of the equilateral triangle, and of the regular tetrahedron. Do these results agree with our earlier findings?

--------------------End of assignment--------------------

Definitions:

1. Transposition (also known as a swap): A transposition(or swap) is a 2-cycle. Ex: $(4 7)$ is a swap, $(1 2)(3 4)$is not a swap.

Theorems:

1. Formula expressing any cycle as a product of transpositions: Any cycle can be written as a product of swaps; Any permutation in $S_n$ can be written as a product of swaps.$i_1, i_2, \cdots, i_{l-1}, i_{l}) = (i_1, i_l)(i_1, i_{l-1})\cdots(i_1, i_2)$.Ex: Write $(3 8 2 4 6)$ as a product of swaps:$(3 6)(3 4)(3 2)(3 8)$ # Theorem concerning generation of $S_n$ by transpositions ("Any subgroup of $S_n$ which contains all of the transpositions is..."): The subgroup of $S_n$ generated by the set of all swaps in $S_n$ is $S_n$. (I.e. any subgroup of $S_n$ that contains every transposition is all of $S_n$.)