# 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 $F$ ${\mathrm{Mat}}_{n}\left(F\right)$ the ring of all $n×n$ matrices in $F$ ${\mathrm{GL}}_{n}\left(F\right)$ the multiplicative group ${\left({\mathrm{Mat}}_{n}\left(F\right)\right)}^{*}$ det the homomorphism $\phantom{\rule{0.3em}{0ex}}{\mathrm{GL}}_{n}\left(F\right)\to {F}^{*}\phantom{\rule{0.3em}{0ex}}$ given by taking the determinant of a matrix ${\mathrm{SL}}_{n}\left(F\right)$ the kernel of the homomorphism det ${\nu }_{n}$ the homomorphism $\phantom{\rule{0.3em}{0ex}}{F}^{*}\to {\mathrm{GL}}_{n}\left(F\right)\phantom{\rule{0.3em}{0ex}}$ given by $\phantom{\rule{0.3em}{0ex}}a↦a·{1}_{n}\phantom{\rule{0.3em}{0ex}},\phantom{\rule{0.3em}{0ex}}$ (${1}_{n}$ the identity) ${\mathrm{PGL}}_{n}\left(F\right)$ the quotient group $\phantom{\rule{0.3em}{0ex}}{\mathrm{GL}}_{n}\left(F\right)⁄\mathrm{Im}\left({\nu }_{n}\right)\phantom{\rule{0.3em}{0ex}}$

1. For each infinite field $F$ and each integer $n\ge 2$ provide an example of an infinite subgroup of the group ${\mathrm{GL}}_{n}\left(F\right)$ of invertible $n×n$ 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 ${\mathrm{GL}}_{n}\left(F\right)$.

2. Let $R$ be a commutative ring and $I$ an ideal in $R$. One says that two matrices $A$ and $B$ in ${\mathrm{Mat}}_{n}\left(R\right)$ 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 ${\mathrm{Mat}}_{n}\left(R\right)$, and describe the quotient ring ${\mathrm{Mat}}_{n}\left(R\right)⁄{J}_{n}$.

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

4. Let ${\mathbf{F}}_{3}$ denote the field of $3$ elements. Observe that the order of the group ${\mathrm{GL}}_{2}\left({\mathbf{F}}_{3}\right)$ is $48$ and that the groups ${\mathrm{SL}}_{2}\left({\mathbf{F}}_{3}\right)$ and ${\mathrm{PGL}}_{2}\left({\mathbf{F}}_{3}\right)$ 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.

5. 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 $\mathrm{Hs}\left(V\right)$ is the set $V×{V}^{*}×F$ with group law given by $\left({v}_{1},{f}_{1},{t}_{1}\right)*\left({v}_{2},{f}_{2},{t}_{2}\right)\phantom{\rule{0.6em}{0ex}}=\phantom{\rule{0.6em}{0ex}}\left({v}_{1}+{v}_{2},{f}_{1}+{f}_{2},{t}_{1}+{t}_{2}+{f}_{2}\left({v}_{1}\right)\right)\phantom{\rule{0.6em}{0ex}}\text{.}$

1. Show that the center1 $C$ of $\mathrm{Hs}\left(V\right)$ is the set $\left\{\left(0,0,t\right)\in \mathrm{Hs}\left(V\right)\phantom{\rule{0.4em}{0ex}}|\phantom{\rule{0.4em}{0ex}}t\in F\right\}\phantom{\rule{0.6em}{0ex}}\text{.}$

2. Let $H$ denote the set $\left\{\left(0,f,0\right)\in \mathrm{Hs}\left(V\right)\phantom{\rule{0.4em}{0ex}}|\phantom{\rule{0.4em}{0ex}}f\in {V}^{*}\right\}\phantom{\rule{0.6em}{0ex}}\text{.}$ Show that $H$ is a subgroup of $\mathrm{Hs}\left(V\right)$ that is isomorphic to the additive group of ${V}^{*}$.

3. Let $N$ denote the set $\left\{\left(v,0,t\right)\in \mathrm{Hs}\left(V\right)\phantom{\rule{0.4em}{0ex}}|\phantom{\rule{0.4em}{0ex}}v\in V,\phantom{\rule{0.6em}{0ex}}t\in F\right\}$ Show that $N$ is a normal subgroup of $\mathrm{Hs}\left(V\right)$.

4. Show that the quotient $\mathrm{Hs}\left(V\right)⁄N$ is isomorphic to ${V}^{*}$.

5. Since $N$ is normal in $\mathrm{Hs}\left(V\right)$, the subgroup $H$ conjugates $N$ to itself, i.e., one has $hn{h}^{-1}\in N$ 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

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