Tutorial questions

1. (Easy)

2. (1)
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   (*) Return all upper-triangular matrices of $\mathrm{GL}(3,q)$, for $q\geq 2$ a prime power. Check, for small $q$, that this is a group of order $q^{3}(q-1)^{3}$.

3. (2)
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   Compute a Sylow $2$-subgroup of $\mathrm{GL}(3,5)$, and return all its abelian subgroups of exponent four.

4. (Medium)

5. (3)
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   Show that $\mathrm{GL}(4,2)$ contains a subgroup isomorphic to $\mathrm{SL}(3,2)$.

   Hint: Look at the orders.

6. (4)
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   (*) Construct the holomorph of the elementary abelian group of order $25$. Recall that the holomorph of a group $G$ is the semidirect product of the group with its automorphism group.

7. (Hard)

8. (5)
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   (*) Show the inner automorphism group of $\mathrm{SL}(2,7)$ is naturally a subgroup of the inner automorphism group of $\mathrm{GL}(2,7)$. Construct such an inclusion with GAP.

   Hint: Write the corresponding group diagrams, and look at the centres.

9. (6)
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   (*) Define the action of $\mathrm{GL}(n,q)$ on subspaces of its natural module (the vector space of dimension $n$ over the field of $q$ elements). Compute the corresponding stabilisers, up to conjugacy, in the case of $n=3$ and $q=4$. Does every subgroup of $\mathrm{GL}(3,4)$ arise as a stabiliser?

10. (7)
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    Construct the homomorphism $\operatorname{Alt}_{6}\to\operatorname{Aut}(\operatorname{Alt}_{6})$, arising from the conjugation action.

    1. (a)
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       Show that this map is injective, and its image is a normal subgroup.

    2. (b)
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       Show that $\operatorname{Aut}(\operatorname{Alt}_{6})/\operatorname{Alt}_{6}$ is isomorphic to $C_{2}\times C_{2}$, the direct product of two cyclic groups of order two. Conclude that there must be exactly five non-trivial normal subgroups in $\operatorname{Aut}(\operatorname{Alt}_{6})$, without using a subgroup function like NormalSubgroups to compute them.

       Hint: Use that $\operatorname{Alt}_{6}$ is simple and the isomorphism theorems.

    3. (c)
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       Show that there is a unique subgroup of $\operatorname{Aut}(\operatorname{Alt}_{6})$ isomorphic to $\operatorname{Sym}_{6}$, and construct such a subgroup by looking at the image of the conjugation action homomorphism.

More exercises

1. (8)
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   Show that $\mathrm{SL}(3,7)$ has a unique non-trivial proper normal subgroup, and determine its structure. Do you recognise this subgroup?

2. (9)
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   Exhibit two non-equivalent transitive actions of $\mathrm{SL}(3,2)$ on a set of seven elements. Can you recognise these actions? Are there more?

3. (10)
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   Compute, up to conjugacy, the fixed subspaces of the order-$2$ elements in $\mathrm{GL}(3,4)$ acting on its natural module.

4. (11)
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   Write a function such that, given a subgroup $H$ of $\mathrm{GL}(n,q)$, it returns the list of $H$-stable $2$-dimensional subspaces. Evaluate this function for $n=2$, $2\leq q\leq 19$, and $H$ the Sylow $p$-subgroup of $\mathrm{GL}(n,q)$, with $p$ the underlying characteristic.

5. (12)
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   Consider the group $\mathrm{GL}(3,q)$, for $2\leq q\leq 9$.

   1. (a)
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      Compute the subgroup $U$ of strictly upper-triangular matrices of $\mathrm{GL}(3,q)$ (with $1$s along the diagonal). What can you say about this group?

   2. (b)
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      Show that the normaliser of such a subgroup, say $B$, is exactly the subgroup of upper-triangular matrices. This $B$ is called a Borel subgroup of $\mathrm{GL}(3,q)$.

   3. (c)
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      Compute all subgroups of $\mathrm{GL}(3,q)$ containing $B$. These are the parabolic subgroups of $\mathrm{GL}(3,q)$ containing $B$.

   4. (d)
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      Let $p$ be the underlying characteristic. Show that the previous family is in bijection with the $p$-subgroups $R$ of $U$ for which the $p$-core of the normaliser $N_{\mathrm{GL}(n_{q})}(R)$ is exactly $R$.

   5. (e)
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      (Hard) Show that the previous bijection can be naturally made order-reversing.

6. (13)
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   Let $H=\mathrm{GL}(2,3)\times\mathrm{GL}(2,3)$ be embedded in $\mathrm{GL}(4,3)$ in the usual way by block matrices along the diagonal, and consider its natural action on the $4$-dimensional vector space over the field of three elements, with the canonical basis given by $e_{1},\ldots,e_{4}$. Show that the only $H$-invariant subspaces are $0$, $A=\langle e_{1},e_{2}\rangle$, $B=\langle e_{3},e_{4}\rangle$, and $A\oplus B$.