Univ at Albany: Math: W. F. Hammond: Courses: Math 520A

Math 520A Written Assignment No. 1

due Wednesday, February 14, 2007

Directions. This assignment should be typeset. If you make use of a reference other than class notes, you must properly cite that use. You may not seek help from others.

Notation: Let

F

be a field. The following notations will be used.

$F^{*}$		the multiplicative group of $F$
${Mat}_{n} (F)$		the ring of all $n \times n$ matrices in $F$
${GL}_{n} (F)$		the multiplicative group ${({Mat}_{n} (F))}^{*}$
det		the homomorphism ${GL}_{n} (F) \to F^{}$ given by taking the determinant* of a matrix
${SL}_{n} (F)$		the kernel of the homomorphism det
$ν_{n}$		the homomorphism $F^{*} \to {GL}_{n} (F)$ given by $a \mapsto a \cdot 1_{n},$ ( $1_{n}$ the identity)
${PGL}_{n} (F)$		the quotient group ${GL}_{n} (F) ⁄ Im (ν_{n})$

For each infinite field $F$ and each integer $n \geq 2$ provide an example of an infinite subgroup of the group ${GL}_{n} (F)$ of invertible $n \times n$ matrices in $F$ that both contains a finite subgroup isomorphic to the group of permutations of the $n$ coordinate axes of $F^{n}$ and is not a normal subgroup of ${GL}_{n} (F)$ .

Let $R$ be a commutative ring and $I$ an ideal in $R$ . One says that two matrices $A$ and $B$ in ${Mat}_{n} (R)$ are congruent modulo $I$ if the difference matrix $A - B$ has entries in $I$ . Show that the set of matrices congruent to $0$ modulo $I$ is a two-sided ideal $J_{n}$ in ${Mat}_{n} (R)$ , and describe the quotient ring ${Mat}_{n} (R) ⁄ J_{n}$ .

Determine the number of isomorphism classes among commutative rings (having $4$ elements) of the form $F_{2} [t] ⁄ (t^{2} + a t + b) F_{2} [t]$ where $F_{2}$ is the field with $2$ elements, $F_{2} [t]$ the ring of polynomials with coefficients in $F_{2}$ , and $a, b \in F_{2}$ .

Let $F_{3}$ denote the field of $3$ elements. Observe that the order of the group ${GL}_{2} (F_{3})$ is $48$ and that the groups ${SL}_{2} (F_{3})$ and ${PGL}_{2} (F_{3})$ both have order $24$ . The group $S_{4}$ of all permutations of a set of $4$ elements also has order $24$ . Determine which, if any, of these three groups of order $24$ are isomorphic.

Let $F$ be a field, and let $V$ be a finite-dimensional vector space over $F$ . $V^{*}$ will denote the dual space of $V$ . The Heisenberg group $Hs (V)$ is the set $V \times V^{*} \times F$ with group law given by $(v_{1}, f_{1}, t_{1}) * (v_{2}, f_{2}, t_{2}) = (v_{1} + v_{2}, f_{1} + f_{2}, t_{1} + t_{2} + f_{2} (v_{1})) .$

Show that the center¹ $C$ of $Hs (V)$ is the set $\{(0, 0, t) \in Hs (V)| t \in F\} .$
Let $H$ denote the set $\{(0, f, 0) \in Hs (V)| f \in V^{*}\} .$ Show that $H$ is a subgroup of $Hs (V)$ that is isomorphic to the additive group of $V^{*}$ .
Let $N$ denote the set $\{(v, 0, t) \in Hs (V)| v \in V, t \in F\}$ Show that $N$ is a normal subgroup of $Hs (V)$ .
Show that the quotient $Hs (V) ⁄ N$ is isomorphic to $V^{*}$ .
Since $N$ is normal in $Hs (V)$ , the subgroup $H$ conjugates $N$ to itself, i.e., one has $h n h^{- 1} \in N$ for all $n \in N$ and all $h \in H$ , and this provides an action of $H$ on $N$ . Describe the action of (the additive group of ) $V^{*}$ on $N$ that corresponds via the isomorphism of $H$ with $V^{*}$ to this action of $H$ on $N$ .

Footnotes

* Definition. The center of a group is the subset of the group consisting of those elements that commute with every element of the group.