189-456A: Algebra 3

Assignment 3

Due: Monday, October 21.

1. Let $X_1$ and $X_2$ be two $G$-sets for which every $g\in G$ acting on $X_1$ and on $X_2$ has the same number of fixed points, i.e., $FP_{X_1}(g) = FP_{X_2)(g)$ for all $g\in G$. Are the $G$-sets $X_1$ and $X_2$ necessarily isomorphic? If the answer is yes, prove it, and provide a counterexample if the answer is no. (Hint: reflect on the examples of group actions you have seen in the previous two assignments.)

2. A $G$-set $X$ is said to be doubly transitive if, for all $x,y, x',y'\in X$ with $x\ne y$ and $x'\ne y'$, there is a $g\in G$ satisfying $g x=x'$ and $gy =y'$. Show that $X$ is doubly transitive if and only if $$ \frac{1}{\# G} \sum_{g\in G} {\rm FP}_X(g)^2 = 2.$$ (Hint: Apply Burnside's Lemma to the Cartesian product $X\times X$.)

3. Give a formula (as a function of $t$) for the number of essentially distinct ways of coloring the four faces of a regular tetrahedron with $t$ colors. (I.e., two colorings that just differ by a rotational symmetry of the regular tetrahedron are regarded as the same.) Use your formula to write down this number for $t=1, 2,3, \ldots, 5$. Explain the number you obtained for $t=2$ by a direct combinatorial argument.

4. Let $S_6$ be the symmetric group on $6$ elements. Show that $S_6$ has two conjugacy classes of elements of order $3$, and calculate their cardinalities. What do you observe?
Show that the exceptional outer automorphism of $S_6$ that you constructed at the end of Assignment $1$ interchanges the elements in these two conjugacy classes.

Remark. It is an instructive exercise to list the conjugacy classes and the class equation for $S_6$ and compute the fixed points for $S_6$ acting on $6$ elements in the ``obvious" and ``non-obvious" way. It is a challenging application of the ideas we have covered, and recommended as a way of preparing for the midterm exam.

5. Let $p$ be a prime and let $G=S_p$ be the group of all permutations on $X=\{1,2,\ldots,p\}$. Show that there are $(p-1)!$ elements of $G$ of order $p$ and $(p-2)!$ subgroups of $G$ of order $p$. Use this and Sylow's theorem to give a proof of Wilson's theorem (the assertion in elementary number theory that $(p-1)!$ is congruent to $-1$ modulo $p$).

6. Keeping the notations of question $5$, show that the normaliser $N$ of any Sylow $p$-subgroup of $G$ is a group of cardinality $p(p-1)$. After identifying $X$ with the field $F$ with $p$ elements, show that the normaliser of the Sylow $p$-subgroup generated by the $p$-cycle $(1 2\cdots p)$ is the group of affine linear transformations $x\mapsto ax+b$ with $a\in F^\times$ and $b\in F$.

7. Let $F =\{0,1\}$ be the finite field with $2$ elements and let $G = {\rm GL}_3(F)$ be the finite group of cardinality $168 = 2^3 \cdot 3 \cdot 7$ which already made a cameo appearance in the previous assignment. (a) Using the Sylow theorems, say how many Sylow $7$-groups $G$ contains, and how many elements of order $7$.

(b) Show that the elements $\left(\begin{array}{ccc} 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 1 \end{array}\right)$ and $\left(\begin{array}{ccc} 0 & 0 & 1 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{array}\right)$ are elements of order $7$ in $G$ that are not conjugate to each other.

( Hint: To check that these matrices are of order $7$, you could multiply each matrix with itself $6$ times to check that you get the identity matrix. You should resist this urge: it is less painful to realise instead each group element as a permutation of the $7$ non-zero column vectors in $F^3$. To show that the two matrices are not conjugate to each other, you might want to dust off your linear algebra notes from a previous year and remind yourselves of the properties of the trace of a matrix.)

8. Show that the group $G$ of question 7 has a subgroup of index $7$ which is isomorphic to the permutation group $S_4$. (Hint: let $H$ be the subgroup of $G$ that fixes a non-zero vector $v\in F^3$, and consider its action on the set of two-dimensional subspaces in $F^3$ that do not contain $v$.)

Remark. There is a dual way of approaching this question, by letting $H^*$ be the subgroup of $G$ that fixes a two-dimensional subspace $W\subset F^3$, and considering its action on the set of vectors in $F^3$ that do not belong to $W$. This group is isomorphic to $H$ but not conjugate to it.

9. Use the results you've obtained in question 8 to describe the Sylow $2$-subgroups of $G$ (which abstract group are they isomorphic to?) and to count how many there are. Same question for the Sylow $3$-subgroups.

10. Let $M_n(F)$ denote the ring of $n\times n$ matrices with coefficients in a field $F$, and let $M$ be an element of $M_n(F)$ whose characteristic polynomial is irreducible over $F$. Show that the subring of $M_n(F)$ generated by $F$ and by $M$ is a field. (I.e., a commutative subring in which every non-zero element has a multiplicative inverse.)