189-456A: Algebra 3

Assignment 2

Due: Wednesday, October 2.

1. Let $G$ be a finite group and let $p$ be the smallest prime dividing the cardinality of $G$. Show that any subgroup $H$ of index $p$ in $G$ is necessarily normal. (Hint: consider the action of $G$ on $G/H$ and think about the kernel of the corresponding homomorphism from $G$ to the permutation group on $G/H$.)

2. Let $G$ be a group of cardinality $15$. Show that the Sylow $3$-subgroup and the Sylow $5$ subgroup of $G$ are both unique, hence normal. Use this to conclude that $G$ must be cyclic, i.e., that it contains an element of order $15$.

3. List the possible orders of elements of the permutation group $S_5$ on $5$ elements. (Hint: how do you read off the order of a permutation from its cycle shape?) Use this along with question 2 to conclude that $S_5$ has no subgroup of order $15$, even though its cardinality, $120$, is a multiple of $15$. Show that $S_5$ cannot act transitively on a set of size $8$, but can be made to act transitively on a set of size $d$ for any divisor $d>8$ of $120$.

Remark (not on the assignment): This example shows that the laws governing the cardinalities of possible subgroups of a group $G$ can be a little bit complicated, and that the hypothesis on the cardinality of a subgroup being of $p$-power order cannot be readily relaxed in the Sylow theorems.

4. Let $F$ be any field and let $G:= {\rm GL}_2(F)$ be the group of invertible $2\times 2$ matrices with entries in $F$.

(a) Show that the determinant gives a surjective homomorphism from $G$ to $F^\times$. The kernel of this homomorphism is called the special linear group, and denoted ${\rm SL}_2(F)$.

(b) Show that the group $Z=F^\times \subset G$ of scalar matrices (i.e., scalar multiples of the $2\times 2$ identity matrix) is a normal subgroup of $G$. The quotient $G/Z$ is called the projective linear group, and is denoted ${\rm PGL}_2(F)$.

5. Let $G$ be the group ${\rm SL}_2(F)$ described in question 4.

(a) Show that the group consisting of the identity matrix and its negative is a normal subgroup of $G$. The quotient ${\rm SL}_2(F)/\{\pm 1\}$ is called the projective special linear group, and is denoted ${\rm PSL}_2(F)$.

(b) Compute the cardinality of ${\rm PSL}_2(F_p)$ where $F_p$ denotes the finite field with $p$ elements.

Motivational remark (not part of the assignment). In group theory, the finite groups that contain no proper non-trivial normal subgroups are said to be simple. Simple groups admit no non-trivial surjective homomorphisms to any group of smaller cardinality (since the kernel of such a homomorphism would have to be a non-trivial normal subgroup, and there are none.) They are to be envisaged as the basic constituents in finite group theory, like the elements of the periodic table in chemistry, or the fundamental particles in physics. The groups ${\rm PSL}_2(F_p)$ turn out to be simple for all $p=5,7,\ldots$. They are an example of an infinite family of simple groups, the alternating groups $A_n$ being another instance that you might have seen in your previous group theory class. The smallest non-abelian simple group is the alternating group $A_5$, which has cardinality $60$. It turns out that the next-smallest non-abelian simple group is the group ${\rm PSL}_2(F_7)$ of cardinality $168$. The full classification of the finite simple groups is a monumental achievement which, according to Wikipedia, "consists of tens of thousands of pages in several hundred journal articles written by about 100 authors, published mostly between 1955 and 2004."

6. Let ${\bf F}_p$ be the field with $p$ elements and let $G={\rm GL}_n({\bf F}_p)$ be the group of invertible $n\times n$ matrices with entries in this field.

(a) What is the cardinality of $G$ and of its Sylow $p$-subgroup?

(b) Can you describe the Sylow $p$-subgroup of $G$ concretely? (Hint: think about the set of upper-triangular invertible matrices in $G$.)

7. Let $F_2=\{0,1\}$ denote the field with $2$ elements, let $V$ be a $3$-dimensional vector space over $F_2$, and let $G={\rm Aut}(V)$ be the group of invertible linear transformations from $V$ to itself. Show that $G$ is isomorphic to the group ${\rm GL}_3(F_2)$ of invertible $3\times 3$ matrices with entries in $F_2$, and therefore has cardinality $168$. (Hint: this is really a basic linear algebra question disguised as a group theory question.)

Remark (not part of the assignment). You will note that $168$ has already made an appearance in the motivational remark after question 5. This is no accident. The groups ${\rm PSL}_2(F_7)$ and ${\rm GL}_3(F_2)$ turn out to be isomorphic, although this is very far from being obvious! Isomorphisms of this kind are among the notable charms of the subject.

8. Keeping the same notations as in question 7, recall that the dual space of $V$, denoted $V^*$, is the set of all $F_2$-valued linear functionals on $V$. (A linear functional on $V$ is an $F$- linear transformation from $V$ to $F$.) Let $X := V-\{0\}$ be the set of non-zero vectors in $V$, and let $X^* := V^*-\{0\}$ be the set of non-zero vectors in $V^*$. Show that both $X$ and $X^*$ are transitive $G$-sets of cardinality $7$, after defining the action of $G$ on $X^*$ by the rule $$ g \ell (v) = \ell(g^{-1} v), \qquad \mbox{ for all } g\in G, \ \ \ \ell\in V^*, \ \ \ v \in V.$$ Show that $X^*$ is isomorphic as a $G$-set to the set of two-dimensional subspaces of $V$. (Hint: don't overthink this one, and just try to assign to each linear functional a two-1dimensional subspace, and to every two-dimensional subspace a linear functional, following recipes that are mutually inverse to each other and are so natural and effortless that they have no choice but to be compatible with the action of $G$.)

9. We keep the same notations as in questions 7 and 8. Given a non-zero vector $v\in V$ and a two-dimensional subspace $W\subset V$, show that there is a $g\in G$ satisfying $g W = W$ and $gv\ne v$. Use this, along with question 8, to conclude that $X$ and $X^\ast$ are not isomorphic to each other as $G$-sets.

Remark (not for the assignment). When you first encountered duality in linear algebra, you were told that the dual of an $n$-dimensional vector space is also an $n$-dimensional vector space, and hence that a finite dimensional vector space $V$ is abstractly isomorphic to its dual $V^*$ as vector spaces. But before you got to comfortable with this fact, you were sternly warned that there is no ``canonical", natural isomorphism between $V$ and $V^*$, and hence that it is best to keep the two notions seperate in your mind. Problem 9 makes the sentiment that there is no natural identification between $V$ and $V^*$ more precise, by showing that these two vector spaces are not isomorphic as $G$-sets, where $G={\rm Aut}(V)$.

10. Show that the permutation group $S_6$ contains two conjugacy classes of elements of order $6$, and show that they have the same cardinality (by computing and comparing).

Remark (not for the assignment). You will recall that we constructed an exceptional, non-inner automorphism of $S_6$ in question 10 of the previous assignment. If you want to challenge yourself, you might want to try to show that this exceptional automorphism interchanges the elements in the two conjugacy classes of order $6$, thereby inducing a bijection between them. Alternately, you may want to count the sizes of conjugacy classes in $S_6$ of elements of all possible orders ($1$, $2$, $3$, $4$, $5$ and $6$), and observe that these cardinalities satisfy a remarkable regularity, which can be viewed as an indirect manifestation of the existence of this outer automorphism.