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Section Permutations and Symmetric Groups
Worksheet Part 1: Permutation Computations
The goal of these problems is to practice computing in permutation groups and determine the order of a cycle.
1.
Let
\(\alpha=(1\;2\;5)(4\;6\;7)\) and
\(\beta=(1\;3\;4\;7\;2)(5\;6)\text{.}\) Write each of the following in disjoint cycle form. Remember to work from right to left!
(a)
(b)
(c)
2.
Compute the order of each permutation below.
(a)
(b)
(c)
(d)
(e)
(f)
Worksheet Part 2: Cycle Inverses and Decomposition Into Transpositions
In this part of the activity you will learn how to quickly compute the inverse of a cycle and investigate cycle types and parity.
1. In the first activity you worked out that the order of an
\(m\) -cycle is
\(m\text{.}\) From this, you can deduce that the inverse of an
\(m\) -cycle is which power of that cycle between
\(1\) and
\(m\text{?}\)
2. If you didn’t already on the first worksheet, compute the power you found in the last exercise for the permutations
\((1\;5\;2)\) and
\((2\;5\;3\;4)\text{.}\)
Solution .
\((1\;5\;2)^3 = (1\;2\;5)\) and
\((2\;5\;3\;4)^4 = (2\;4\;3\;5)\text{;}\) in general we can find the inverse of an
\(m\) -cycle by writing the first element and then writing the remaining elements in reverse order.
3. Give a formula for
\((a_1\; a_2\; \dots\; a_m)^{-1}\text{.}\)
Solution .
See the previous solution:
\((a_1\; a_2\; \dots\; a_m)^{-1} = (a_1\; a_m\; a_{m-1}\; \dots\; a_2)\text{.}\)
Definition 44 . Cycle Type of a Permutation.
Two permutations in
\(S_n\) have the same
cycle type if they have the same number of cycles of each length in their unique decomposition into disjoint cycles.
4. Do permutations with the same cycle type have the same parity?
Solution .
Yes, because the parity of each length of cycle is fixed.
5. Fill in the chart below for
\(S_4\text{.}\)
Table 45. Cycle Types in \(S_4\)
example element
parity
# elements
order
Solution .
Table 46. Cycle Types in \(S_4 - Solution\)
example element
\(e\) \((12)\) \((123)\) \((1234)\) \((12)(34)\)
parity
even
odd
even
odd
even
# elements
1
6
8
6
3
order
1
2
3
4
2
