MTH3003 Weekly Problems 7

Original Documents: Problem Sheet / My Handwritten Solutions / [[mth3003 weekly problem sheet 7 solutions.pdf|Provided Solutions]]

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7.1. Signatures in $S_{10}$ and Membership of $A_{10}$

Question

Recall from lectures that the signature function $\sigma :S_n\to \{1,-1\}$ is used to define the alternating group $A_n$.

Calculate the signatures of the following permutations in $S_{10}$, and hence determine whether or not they lie in the alternating group $A_{10}$.

1. $g=(614)(5327)(8910)$
2. $h=(12)(34)(56)(78)$
3. $g^{-1}$
4. $gh$
5. $g^{-1}hg$

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7.2. Alternating Group Membership from Cycle Shapes

Question

Determine whether or not permutations with each of the following cycle shapes lie in some alternating group $A_n$.

1. Cycle shape $(3,2,2,2)$
2. Cycle shape $(7,3,3,2,2)$
3. Cycle shape $(5,5,5)$
4. Cycle shape $(2,2,2,2,2)$

You may assume that knowing the cycle shape is enough to determine the signature.

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7.3. Elements of $A_3$ via Cycle Shapes

Question

Without looking at your notes, try to list all the elements in the alternating group $A_3$ by first listing all the cycle shapes that can occur in $A_3$.

1. List all possible cycle shapes in $S_3$.
2. For each cycle shape, compute the signature to decide whether it can occur in $A_3$.
3. For each cycle shape that does occur in $A_3$, write down all permutations with that cycle shape.

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7.4. Cycle Shapes in $S_5$ and Which Lie in $A_5$

Question

List all the cycle shapes that can occur in $S_5$ and determine which of these lie in the alternating group $A_5$.

Hint. Think of an arbitrary permutation in $S_5$, written as a product of disjoint cycles (including $1$-cycles), with the cycles ordered by decreasing length. Removing the brackets, it will look like a row of the symbols $1,2,3,4,5$ in some order. Now put the brackets back in, starting from the largest cycle. Consider in turn the cases where the largest cycle has length $1,2,3,4,$ or $5$, and systematically list all resulting cycle shapes.

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7.5. Listing All Elements of $A_5$

Question

Use your answer to Question 7.4 to list all elements of the alternating group $A_5$.

1. For each cycle shape that occurs in $A_5$, list all permutations in $S_5$ with that cycle shape.
2. Be careful not to double-count cycles that differ only by a cyclic rotation of entries, for example $(123)=(312)=(231)$.
3. Check that the total number of elements you obtain is $|S_5|/2=5!/2=60$.

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