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Parity
Integer Parity: The parity of an integer is wether it is even or odd.
Lemma 12.1 If L, M, N, P are lines, then there exist lines Q, R so that σP σN σM σL = σQ σR .
Proof: Let β = σP σN σM σL It follows from Proposition 11.2 that we may choose lines N 0 , M 0
so that σN 0 σM 0 = σN σM and so that M 0 intersects L. Now we have
β = σP σN σM σL = σP σN 0 σM 0 σL
Let x be a point where the lines M 0 and L intersect. Now, it follows from Proposition 11.2
that we may choose lines P 0 , N 00 so that σP 0 σN 00 = σP σN 0 and so that N 00 contains the point
x. Now we have
β = σP σN 0 σM 0 σL = σP 0 σN 00 σM 0 σL
Now N 00 , M 0 , L all contain the point x. It now follows from Proposition 11.2 that we may
choose a line N 000 containing x so that σN 000 σL = σN 00 σM 0 . Now we have
β = σP 0 σN 00 σM 0 σL = σP 0 σN 000 σL σL = σP 0 σN 000
which completes the proof.
Parity of Isometries: We say that an isometry is even if it is the identity, a translation,
or rotation and is odd if it is a reflection or glide reflection. Note that by the above corollary
every isometry is either even or odd, but not both.
Proposition 12.2 Let β be an isometry which is a product of reflections. Then β is even if
it is a product of an even number of reflections and odd if it is a product of an odd number
of reflections.
Proof: It follows from the above lemma that every product of q reflections can be reduced
to a product of q 0 reflections where q 0 has the same parity as q and q ≤ 3. If q is odd then
we have a reflection or a glide reflection and if q is even we have a rotation or translation.
Corollary 12.3 If α, β are isometries, then αβ is even if α, β are both even or both odd,
and otherwise αβ is odd.
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Proof: Suppose α can be expressed as a product of a transpositions and β can be expressed
as a product of b transpositions. Then we can express αβ as a product of a+b transpositions,
and the result follows immediately from this.
Theorem 12.4 The even isometries (rotations and translations) form a subgroup of the
group of isometries.
Proof: Let H be the set of rotations and translations.
(identity) The identity ι is a translation by 0, so it is in S.
(inverses) The inverse of the translation τv is τ−v and the inverse of the rotation ρx,θ is
ρx,−θ . Thus, β ∈ S implies β −1 ∈ S as desired.
(closure) Let α, β ∈ S and assume that α can be expressed as a product of a reflections
and β as a product of b reflections. Then αβ can be written as a product of a + b
reflections. Since a, b are both even (as these are even isometries), we have that a + b
is even, so αβ is and even isometry and αβ ∈ S.
Parity of Permutations: We say that a permutation π is odd if it can be written as a
product of an odd number of transpositions and even if it can be written as an even number
of transpositions.
Proposition 12.5 Every permutation is even or odd, but not both.
Sketch of Proof: Write the numbers 1, 2, . . . , n in sequence in a column and construct a graph
on these points by adding a directed edge from i to j whenever i < j. For any permutation
δ ∈ Sn , we claim that the parity of δ is the parity of the number of edges which are directed
upward after the permutation δ is applied to our points. To see this, we imagine performing
a sequence of transpositions to these points. Each time we take two of our points, say the
ones presently in positions i, j with i < j and we interchange them. Now, there are j − i − 1
edges between i and i + 1, i + 2, . . . , j − 1 which flip orientation and j − i − 1 edges between
j and i + 1, i + 2, . . . , j − 1 which flip orientation and one extra edge between i and j which
flips orientation. It follows that the number of edges which flip orientation is always odd.
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Alternating Group: We let An denote the set of even permutations in Sn .
Theorem 12.6 An is a subgroup of Sn for every positive integer n.
Proof: Homework.
Group
Permutations
Isometries
Generated by
transpositions
reflections
Even/Odd
Even subgroup
products of an even/odd # of generators
alternating group
translations & rotations