Tutorial questions

1. (Easy)

2. (1)
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   (*) Let $G$ be the group generated by $(1,2,3,4)$, $(5,6,7,8)$, $(1,5)(2,6)(3,7)(4,8)$. Show the following:

   1. (a)
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      $G$ is a group of order $32$.

   2. (b)
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      $G$ is not abelian.

   3. (c)
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      The centre of $G$ is cyclic of order four.

   4. (d)
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      The derived subgroup of $G$ has index $8$.

3. (2)
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   Show that $D_{6}$ is isomorphic to the symmetric group on $3$ elements.

4. (3)
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   Show that $\operatorname{Alt}_{4}$ has no subgroup of order six.

5. (Medium)

6. (4)
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   (*) Prove that the group

   $$\langle(1,2,3,\ldots,7),(2,6)(3,4)\rangle$$

   is simple, has order $168$ and acts transitively on $\{1,\dots,7\}$. Can you recognise this group? (Hint: Use StructureDescription).

7. (5)
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   Count the number of subgroups of $\operatorname{Sym}_{5}$ isomorphic to some Dihedral group.

8. (6)
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   Exhibit those divisors $d$ of $5!$ such that there is no subgroup of $\operatorname{Sym}_{5}$ with index $d$. That is, the “converse" of Lagrange theorem is not true in general.

9. (Hard)

10. (7)
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    Find all the transitive subgroups of $\operatorname{Sym}_{7}$ containing at least two elements of order two, and an element of order three.

11. (8)
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    (*) Count the number of non-trivial $2$-subgroups in $\operatorname{Alt}_{5}$. Show that the number of $2$-elements is $16$. Recall that a $p$-element is an element whose order is a power of $p$.

12. (Challenging)

13. (9)
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    Define the action of $\operatorname{Sym}_{n}$ on the set of partitions of $\{1,\ldots,n\}$. Compute the stabilisers and fixed point sets of $n$-cycles, for $3\leq n\leq 8$. Do you find a pattern?

More exercises

1. (11)
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   Let $G$ be the group generated by the permutations $(1,2)(6,11)(8,12)(9,13)$, $(5,13,9)(6,10,11)(7,8,12)$ and $(2,4,3)(5,8,9)(6,10,13)(7,11,12)$. How many elements of $G$ are commutators?

2. (12)
   1. (a)
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      Show that two distinct Sylow $2$-subgroups in $\operatorname{Alt}_{5}$ intersect trivially.

   2. (b)
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      Show that the conjugation action of $\operatorname{Alt}_{5}$ on the set of Sylow $2$-subgroups is double transitive. This means that for any two pairs $(S_{1},S_{2})$ and $(Q_{1},Q_{2})$ of Sylow $2$-subgroups with $S_{1}\neq S_{2}$ and $Q_{1}\neq Q_{2}$, there is an element $g\in\operatorname{Alt}_{5}$ such that $(S_{1},S_{2})^{g}=(Q_{1},Q_{2})^{g}$.

3. (13)
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   Recall that a group action is primitive if it is isomorphic to a coset action over a maximal subgroup. Write a function that computes, up to conjugacy, the primitive subgroups of $\operatorname{Sym}_{n}$ on its natural action.

4. (14)
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   Find all faithful transitive actions of $D_{16}$, the Dihedral group of order $16$.

5. (15)
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   Show that $\operatorname{Alt}_{34}$ and $\operatorname{Sym}_{34}$ are the only primitive groups of degree $34$.