Combinatorics Through Guided Discovery, 2004a

106 6. Groups acting on sets
still hold if one or more of the exponents may be negative?
(c) What would we have to prove to show that the rules still hold?
(d) If the rules hold, give enough of the proof to show that you know how
to do it; otherwise give a counterexample.
• Problem 254. If a finite set of permutations satisfies the closure property is
it a permutation group? (h)
• Problem 255. There are three-dimensional geometric motions of the square
that return it to its original position but move some of the vertices to other
positions. For example, if we flip the square around a diagonal, most of
it moves out of the plane during the flip, but the square ends up in the
same place. Draw a figure like Figure 6.1.1 that shows all the possible
results of such motions, including the ones shown in Figure 6.1.1. Do the
corresponding permutations form a group?
Problem 256. Let σ and ϕ be permutations.
(a) Why must σ ◦ ϕ have an inverse?
(b) Is (σ ◦ ϕ) −1 = σ −1 ϕ −1 ? (Prove or give a counter-example.) (h)
(c) Is (σ ◦ ϕ) −1 = ϕ −1 σ −1 ? (Prove or give a counter-example.)
• Problem 257. Explain why the set of all permutations of four elements is a
permutation group. How many elements does this group have? This group
is called the symmetric group on four letters and is denoted by S 4 .
6.1.3 The symmetric group
In general, the set of all permutations of an n-element set is a group. It is called
the symmetric group on n letters. We don’t have nice geometric descriptions (like
rotations) for all its elements, and it would be inconvenient to have to write down
something like “Let σ(1) = 3, σ(2) = 1, σ(3) = 4, and σ(4) = 1” each time we need
to introduce a new permutation. We introduce a new notation for permutations
that allows us to denote them reasonably compactly and compose them reasonably
quickly. If σ is the permutation of {1, 2, 3, 4} given by σ(1) = 3, σ(2) = 1, σ(3) = 4