189-456A: Algebra 3

Assignment 3

Due: Sunday, October 22.

1. Let $G$ be a finite group and let $X$ and $Y$ be finite $G$-sets. As in class, denote by FP$_X(g)$ the number of fixed points of $g\in G$ on $X$, and likewise for $Y$. The cartesian product $$X\times Y = \{ (x,y), x\in X, Y\in Y \}$$ is naturally a $G$-set under the action $$ g (x,y) = (gx, gy).$$ Show that FP$_{X\times Y}(g) =$ FP$_X(g)\cdot$ FP$_Y(g)$, for all $g\in G$.

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: Use what you proved in question 1.)

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. Recall from Assignment 1 that the permutation group $S_5$ has a subgroup $F_{20}$ of cardinality $20$, and that it acts transitively on the 6 element set $X = S_5/F_{20}$. Describe the conjugacy classes of $S_5$ and their sizes (i.e., compute the class equation of $S_5$). For each conjugacy class $C$, write down ${\rm FP}_X(C)$. Use this to verify the Burnside counting formula for the $S_5$-set $X$.

5. Let $F$ be the finite field with $p$ elements, let $G={\rm GL}_2(F)$ be the group of $2\times 2$ matrices with entries in $F$, acting on the space $F^2$ of column vectors with entries in $F$ by left multiplication. Let $X$ be the set of one-dimensional subspaces of $F^2$. Show that the map ${\rm span}\left(\begin{array}{c} x \\ y \end{array}\right) \mapsto x/y$, (with the natural convention that $a/0 = \infty$) places $X$ in bijection with the projective line $${\bf P}_1(F):= \{0,1,2,\ldots, p-1,\infty\} $$ of cardinality $p+1$. Show that via this identification, the action of $G$ on $X$ is given by $$ \left(\begin{array}{cc} a & b \\ c & d \end{array}\right) t = \frac{at +b}{c t+d}.$$ The function $t \mapsto \frac{at+b}{ct+d}$ is called a Moebius transformation, and the projective line equipped with the action of $G$ by Moebius transformations is a particularly appealing instance of a $G$-set.

6. 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 $p$).

7. Keeping the notations of question $6$, 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 $b\in F$.

8. 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.)

9. Show that the group $G$ of question 8 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.

10. Use the results you've obtained in part 9 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.