189-570A: Higher Algebra I

Assignment 3

Due: Monday, October 28.

1. Recall that for an element $g$ of a group $G$, the centralizer of $g$ in $G$ - the group of elements of $G$ which commute with $G$ - is denoted by $Z(g)$.

a. Show that the function which to $g\in G$ associates the cardinality of $Z(g)$ is the character of a representation of $G$. What is this representation?

b. Using part a, compute the average over $G$ of the cardinality of $Z(g)$ -- the average size of the centralizer of an element of $G$ .

2. Let $F_{20}$ be the Frobenius group of order $20$ in $S_5$. Show that the permutations $(12)$ and $(12)(34)$ of $S_5$ act on the coset space $S_5/F_{20}$ with zero and two fixed points respectively, and therefore that the image of $(12)$ under the homomorphism from $S_5$ to $S_6$ arising from the action of $S_5$ on $S_5/F_{20}$ is a product of three disjoint transpositions, while the image of $(12)(34)$ is a product of two disjoint transpositions. Show that the image of $(1234)$ in $S_6$ is a four-cycle, and use this to give another proof of the fact that $(13)(24)$ acts as a product of two disjoint transpositions in $S_6$.

Let $G = {\bf GL}_3({\bf F}_2)$ be the group of invertible 3 x 3 matrices with entries in the field with two elements. It is a group of cardinality $168$, and is the second smallest non-abelian simple group, after the alternating group $A_5$ of cardinality $60$. It acts naturally by linear transformations on the space $W$ of column vectors of size $3$ with entries in ${\bf F}_2$.

The goal of the following series of exercises is to calculate the character table for G.

Note that you are not obliged to follow the hints if you find another way of approaching the question.

3. Show that $G$ has a unique conjugacy class (to be denoted $2A$) of elements of order $2$ which is of cardinality $21$. (Hint: If $T$ is an element of order two in $G$, show that $T$ has minimal polynomial $(T-1)^2$. Conclude that the linear endomorphism $U=(T-1)$ has a two dimensional kernel and that the image of $U$ is a one-dimensional subspace of kernel$(U)$. Show that the datum $({\rm Image}(U) , {\rm kernel}(U))$ determines $T$ completely.)

4. Show that $G$ has a unique conjugacy class (to be denoted $3A$) of elements of order $3$, which is of cardinality $56$. (Hint: Show that the characteristic polynomial of an element $T$ of order $3$ is equal to $(x+1)(x^2+x+1)$ and that $T$ is completely determined by the following data: the kernel $W_1$ of $T^2+T+1$ and the kernel $W_2$ of $T+1$, yielding a decomposition of $W$ into a direct sum of a two-dimensional and a one dimensional subspace, together with the extra datum of a cyclic permutation of order $3$ on the non-zero vectors in $W_1$.)

5. Show that $G$ has a unique conjugacy class (to be denoted $4A$) of elements of order $4$, which is of cardinality $42$. (Hint: Show that the $21$ distinct Sylow $2$-subgoups of $G$ are isomorphic to the dihedral goup $D_8$ of order $8$, and that the intersection of any two of them is a group of exponent $2$.)

6. Show that G has two distinct conjugacy classes of elements of order $7$ (to be denoted $7A$ and $7B$), which are each of size $24$ and corresponding to the linear transformations having characteristic polynomial $x^3+x^2+1$ and $x^3+x+1$ respectively. (Hint: Show that G has $8$ distinct Sylow $7$-subgroups and hence $48$ elements of order $7$.)

7. Let $X$ be the set of non-zero vectors of $W$ and let $V_6={\rm fct}^0(X,{\mathbb C})$ be the set of complex-valued functions on $X$ of sum $0$. Compute the character attached to $V_6$ and show that $V$ is an irreducible $6$-dimensional representation of G.

8. Let $X$ be the set of Sylow $7$-subgroups of $G$, on which $G$ acts by conjugation, and let $V_7={\rm fct}_0(X,{\mathbb C})$. Compute the character attached to $V_7$ and show that $V$ is an irreducible $7$-dimensional representation of G.

9. Let $W$ be the alternating square of the representation $V_6$ of exercise 7. Show that $W$ decomposes as a direct sum of $V_7$ and a new irreducible representation $V_8$ of dimension $8$. Write down the character of $V_8$.

From the work you have done so far, conclude that the irreducible representations of $G$ are of dimensions $1$, $6$, $7$, $8$, $3$ and $3$ respectively. Of these, only the characters of the two irreducible three-dimensional representations have yet to be computed.

10. Show that $G$ contains a Frobenius subgroup $F_{21}$ of order $21$, which is isomorphic to the group of affine linear transformations of the form $x\mapsto ax+b$ where $a\in \{1,2,4\}\subset {\bf F}_7^\times$ and $b\in {\bf F}_7$. Show that $F_{21}$ admits an irreducible three dimensional representation whose traces belong to ${\bf Q}(\sqrt{-7})$. Compute the character of the $24$-dimensional representation obtained by inducing this representation to $G$, and use it to complete the character table of $G$.