Group Theory and Linear algebra

Semester II 2011

Last updates: 1 September 2011

(1) Week 8: Vocabulary

(2) Week 8: Results

(3) Week 8: Examples and computations

Let $G$ be a group and let $H$ be a subgroup. Define a left coset of $H$, a right coset of $H$ and the index of $H$ in $G$ and give some illustrative examples. | |

Let $G$ be a group and let $H$ be a subgroup. Define $G/H$ and give some illustrative examples. | |

Let $G$ be a group. Define normal subgroup of $G$ and give some illustrative examples. | |

Let $G$ be a group and let $H$ be a normal subgroup. Define the quotient group $G/H$ and give some illustrative examples. |

Let $G$ be a group and let $H$ be a subgroup of $G$. Let $a,b\in G$. Show that $Ha=Hb$ if and only if $a{b}^{-1}\in H$. | |

Let $G$ be a group and let $H$ be a subgroup of $G$. Show that each element of $G$ lies in exactly one coset of $G$. | |

Let $G$ be a group and let $H$ be a subgroup of $G$. Let $a,b\in G$. Show that the function $f:Ha\to Hb$ given by $f\left(ha\right)=hb$ is a bijection. | |

Let $G$ be a group and let $H$ be a subgroup of $G$. Show that $G/H$ is a partition of $G$. | |

Let $G$ be a group and let $H$ be a subgroup of $G$. Let $g\in G$. Show that $gH$ and $H$ have the same number of elements. | |

Let $G$ be a group of finite order and let $H$ be a subgroup of $G$. Show that $\mathrm{Card}\left(H\right)$ divides $\mathrm{Card}\left(G\right)$. | |

Let $G$ be a group of finite order and let $g\in G$. Show that the order of $g$ divides the order of $G$. | |

Let $G$ be a finite group and let $n=\mathrm{Card}\left(G\right)$. Show that if $g\in G$ then ${g}^{n}=1$. | |

Let $p$ be a prime positive integer. Show that if $a$ is an integer which is not a multiple of $p$ then ${a}^{p-1}=1$ mod $p$. | |

Let $p$ be a prime positive integer. Let $G$ be a group of order $p$. Show that $G$ is isomorphic to $\mathbb{Z}/p\mathbb{Z}$. | |

Let $G$ be a group and let $H$ be a subgroup of $G$. Show that $H$ is a normal subgroup of $G$ if and only if $H$ satisfies $$\text{if}\phantom{\rule{0.5em}{0ex}}g\in G\phantom{\rule{2em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}Hg=gH.$$ | |

Let $G$ be a group and let $H$ be a subgroup of $G$. Show that $H$ is a normal subgroup of $G$ if and only if $H$ satisfies $$\text{if}\phantom{\rule{0.5em}{0ex}}g\in G\phantom{\rule{2em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}gH{g}^{-1}=H.$$ | |

Let $G$ be a group and let $H$ be a normal subgroup of $G$. Show that if $a,b\in G$ then $HaHb=Hab$. | |

Let $G$ be a group and let $H$ be a normal subgroup of $G$. Show that $G/H$ with operation given by $\left({g}_{1}H\right)\left({g}_{2}H\right)={g}_{1}{g}_{2}H$ is a group. | |

Let $f:G\to H$ be a group homomorphism. Show that $\mathrm{ker}f$ is a normal subgroup of $G$. | |

Let $f:G\to H$ be a group homomorphism. Show that $\mathrm{im}f$ is a subgroup of $H$. | |

Let $f:G\to H$ be a group homomorphism. Show that $f$ is injective if and only if $\mathrm{ker}f=\left\{1\right\}$. | |

Let $G$ be a group and let $H$ be a normal subgroup
of $G$. Let $f:G\to G/H$ be given by $f\left(g\right)=gH$. Show that
- (a) $f$ is a group homomorphism,
- (b) $\mathrm{ker}f=H$,
- (c) $\mathrm{im}f=G/H$.
| |

Let $f:G\to H$ be a group homomorphism. Show that $G/\mathrm{ker}f\cong \mathrm{im}f.$ |

Let $$A=\left(\begin{array}{cc}0& 1\\ -1& -1\end{array}\right)\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}B=\left(\begin{array}{cc}0& 1\\ -1& 0\end{array}\right)$$ Show that $A$ has order 3, that $B$ has order 4 and that $AB$ has infinite order. | |

Assume that $G$ is a group such that $$\text{if}\phantom{\rule{0.5em}{0ex}}g,h\in G\phantom{\rule{2em}{0ex}}\text{then}\phantom{\rule{1em}{0ex}}{\left(gh\right)}^{2}={g}^{2}{h}^{2}.$$ Show that $G$ is commutative. | |

Decide whether the positive integers is a subgroup of the integers with operation addition. | |

Decide whether the set of permutations which fix 1 is a subgroup of ${S}_{n}$. | |

List all subgroups of $\mathbb{Z}/12\mathbb{Z}$. | |

Let $G$ be a group, let $H$ be a subgroup and let $g\in G$. Show that $gH{g}^{-1}=\left\{gh{g}^{-1}\phantom{\rule{0.3em}{0ex}}\right|\phantom{\rule{0.3em}{0ex}}h\in H\}$ is a subgroup of $G$. | |

Let $G$ be a group and let $g\in G$. Let $f:G\to G$ be given by $f\left(h\right)=gh{g}^{-1}$. Show that $f$ is an isomorphism. | |

Show that ${\mathrm{SO}}_{2}\left(\mathbb{R}\right)$ is isomorphic to ${U}_{1}\left(\u2102\right)$. | |

Show that $(\mathbb{R},+)$ and $({\mathbb{R}}^{\times},\times )$ are not isomorphic. | |

Show that $(\mathbb{Z},+)$ and $(\mathbb{Q},+)$ are not isomorphic. | |

Show that $(\mathbb{Z},+)$ and $({\mathbb{Q}}_{>0},\times )$ are not isomorphic. | |

Show that ${\mathrm{SL}}_{2}\left(\mathbb{Z}\right)$ is a subgroup of ${\mathrm{GL}}_{2}\left(\mathbb{R}\right)$. | |

Find the orders of elements $1,-1,2$ and $i$ in the group ${\u2102}^{\times}=\u2102-\left\{0\right\}$ with operation multiplication. | |

Find the orders of elements in $\mathbb{Z}/6\mathbb{Z}$. | |

Find the subgroups of $\mathbb{Z}/6\mathbb{Z}$. | |

Write the element (345) in ${S}_{5}$ in diagram notation, two line notation, and as a permutation matrix, and determine its order. | |

Write the element (13425) in ${S}_{5}$ in diagram notation, two line notation, and as a permutation matrix, and determine its order. | |

Write the element (13)(24) in ${S}_{5}$ in diagram notation, two line notation, and as a permutation matrix, and determine its order. | |

Write the element (12)(345) in ${S}_{5}$ in diagram notation, two line notation, and as a permutation matrix, and determine its order. | |

Let $n$ be a positive integer. Determine if the group of complex $n$th roots of unity $\{z\in \u2102\phantom{\rule{0.5em}{0ex}}|\phantom{\rule{0.5em}{0ex}}{z}^{n}=1\}$ (with operation multiplication) is a cyclic group. | |

Determine if the rational numbers $\mathbb{Q}$ with operation addition is a cyclic group. | |

Find the order of the element (1,2) in the group $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/8\mathbb{Z}$. | |

Show that the group $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/6\mathbb{Z}$ and the group $\mathbb{Z}/12\mathbb{Z}$ are not isomorphic. | |

Show that the group $\mathbb{Z}\times \mathbb{Z}$ and the group $\mathbb{Q}$ with operation addition are not isomorphic. | |

Let $G$ be a group and let $a,b\in G$.
Assume that $ab=ba$.
- (a) Prove, by induction, that if $n\in {\mathbb{Z}}_{>0}$ then $a{b}^{n}={b}^{n}a$,
- (b) Prove, by induction, that if $n\in {\mathbb{Z}}_{>0}$ then ${a}^{n}{b}^{n}={b}^{n}{a}^{n}$,
- (c) Show that the order of $ab$ divides the least common multiple of the order of $a$ and the order of $b$.
- (d) Show that if $a=\left(12\right)$ and $b=\left(13\right)$ then the order of $ab$ does not divide the least common multiple of the order of $a$ and the order of $b$.
| |

Show that the order of ${\mathrm{GL}}_{2}\left(\mathbb{Z}/2\mathbb{Z}\right)$ is 6. | |

Let $p$ be a prime positive integer. Find the order of the group ${\mathrm{GL}}_{2}\left(\mathbb{Z}/p\mathbb{Z}\right)$. | |

Let $n\in {\mathbb{Z}}_{>0}$ and let $p$ be a prime positive integer. Find the order of the group ${\mathrm{GL}}_{n}\left(\mathbb{Z}/p\mathbb{Z}\right)$. | |

Show that the group $\mathbb{Z}\left[x\right]$ of polynomials with integer coefficients with operation addition is isomorphic to the group ${\mathbb{Q}}_{>0}$ with operation multiplication. | |

Let $G$ be a group with less than 100 elements which has subgroups of orders 10 and 25. Find the order of $G$. | |

Let $G$ be a group and let $H$ and $K$ be subgroups of $G$. Show that $|H\cap K|$ is a common divisor of $\left|H\right|$ and $\left|K\right|$. | |

Let $G$ be a group and let $H$ and $K$ be subgroups of $G$. Assume that $\left|H\right|=7$ and $\left|K\right|=29$. Show that $H\cap K=\left\{1\right\}$. | |

Let $H$ be the subgroup of $G=\mathbb{Z}/6\mathbb{Z}$ generated by 3. Compute the right cosets of $H$ in $G$ and the index $|G:H|$. | |

Let $H$ be the subgroup of $G=\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$ generated by $(1,0)$. Find the order of each element in $G/H$ and identify the group $G/H$. | |

Let $H$ be the subgroup of $G=\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/4\mathbb{Z}$ generated by $(0,2)$. Find the order of each element in $G/H$ and identify the group $G/H$. | |

Let $n\in {\mathbb{Z}}_{\ge 2}$ and define $f:{\mathrm{GL}}_{n}\left(\u2102\right)\to {\mathrm{GL}}_{n}\left(\u2102\right)$ by $f\left(A\right)={A}^{t}$. Determine whether $f$ is a group homomorphism. | |

Let $n\in {\mathbb{Z}}_{\ge 2}$ and define $f:{\mathrm{GL}}_{n}\left(\u2102\right)\to {\mathrm{GL}}_{n}\left(\u2102\right)$ by $f\left(A\right)={\left({A}^{-1}\right)}^{t}$. Determine whether $f$ is a group homomorphism. | |

Let $n\in {\mathbb{Z}}_{\ge 2}$ and define $f:{\mathrm{GL}}_{n}\left(\u2102\right)\to {\mathrm{GL}}_{n}\left(\u2102\right)$ by $f\left(A\right)={A}^{2}$. Determine whether $f$ is a group homomorphism. | |

Let $B$ be the subgroup of ${\mathrm{GL}}_{2}\left(\mathbb{R}\right)$ of upper triangular matrices and let $T$ be the subgroup of ${\mathrm{GL}}_{2}\left(\mathbb{R}\right)$ of diagonal matrices. Let $f:B\to T$ be given by $$f\left(\left(\begin{array}{cc}a& b\\ 0& c\end{array}\right)\right)=\left(\begin{array}{cc}a& 0\\ 0& c\end{array}\right).$$ Show that $f$ is a group homomorphism. Find $N=\mathrm{ker}f$ and identify the quotient $B/N$. | |

Assume $G$ is a cyclic group and let $N$ be a subgroup of $G$. Show that $N$ is a normal subgroup of $G$ and that $G/N$ is a cyclic group. | |

Simplify ${3}^{52}$ mod 53. | |

Suppose that ${2}^{147052}=76511$ mod 147053. What can you conclude about 147053? | |

Show that if $f:G\to H$ is a group homomorphism and ${a}_{1},{a}_{2},\dots ,{a}_{n}\in G$ then $f({a}_{1}{a}_{2}\cdots {a}_{n})=f\left({a}_{1}\right)f\left({a}_{2}\right)\cdots f\left({a}_{n}\right)$. | |

Describe all group homomorphisms $f:\mathbb{Z}\to \mathbb{Z}$. | |

Show that ${\mathrm{SO}}_{n}\left(\mathbb{R}\right)$ is a normal subgroup of ${O}_{n}\left(\mathbb{R}\right)$ by finding a homomorphism $f:{O}_{n}\left(\mathbb{R}\right)\to \{\pm 1\}$ with kernel ${\mathrm{SO}}_{n}\left(\mathbb{R}\right)$. Identify the quotient ${O}_{n}\left(\mathbb{R}\right)/{\mathrm{SO}}_{n}\left(\mathbb{R}\right)$. | |

Show that ${\mathrm{SU}}_{n}\left(\u2102\right)$ is a normal subgroup of ${U}_{n}\left(\u2102\right)$ by finding a homomorphism $f:{U}_{n}\left(\u2102\right)\to {U}_{1}\left(\u2102\right)$ with kernel ${\mathrm{SU}}_{n}\left(\u2102\right)$. Identify the quotient ${U}_{n}\left(\u2102\right)/{\mathrm{SU}}_{n}\left(\u2102\right)$. | |

Let $G$ be a group and let $H$ be a subgroup of $G$. Let $f:G/H\to H\backslash G$ be given by $f\left(aH\right)=H{a}^{-1}$. Show that $f$ is a function and that $f$ is a bijection. | |

Let $G=\mathbb{Z}$ and $H=2\mathbb{Z}$. Compute the cosets of $H$ in $G$ and the index $|G:H|$. | |

Let $G={S}_{3}$ and let $H$ be the subgroup generated by (123). Compute the cosets of $H$ in $G$ and the index $|G:H|$. | |

Let $G={S}_{3}$ and let $H$ be the subgroup generated by (12). Compute the cosets of $H$ in $G$ and the index $|G:H|$. | |

Let $G={\mathrm{GL}}_{2}\left(\mathbb{R}\right)$ and let $H={\mathrm{SL}}_{2}\left(\mathbb{R}\right)$. Compute the cosets of $H$ in $G$ and the index $|G:H|$. | |

Let $G$ be the subgroup of ${\mathrm{GL}}_{2}\left(\mathbb{R}\right)$ given by $$G=\left\{\left(\begin{array}{cc}x& y\\ 0& 1\end{array}\right)\phantom{\rule{0.3em}{0ex}}\right|\phantom{\rule{0.3em}{0ex}}x,y\in \mathbb{R},x>0\}$$ Let $H$ be the subgroup of $G$ given by $$H=\left\{\left(\begin{array}{cc}z& 0\\ 0& 1\end{array}\right)\phantom{\rule{0.3em}{0ex}}\right|\phantom{\rule{0.3em}{0ex}}z\in \mathbb{R},z>0\}.$$ Each element of $G$ can be identified with a point $(x,y)$ of ${\mathbb{R}}^{2}$. Use this to describe the right cosets of $H$ in $G$ geometrically. Do the same for the left cosets of $H$ in $G$. | |

Consider the set $AX=B$ of linear equations where $X$ and $B$ are column vectors, $X$ is the matrix of unknowns, and $A$ the matrix of coefficients. Let $W$ be the subspace of ${\mathbb{R}}^{n}$ which is the set of solutions of the homogeneous equations $AX=0$. Show that the set of solutions of $AX=B$ is either empty or is a coset of $W$ in the group ${\mathbb{R}}^{n}$ (with operation addition). | |

Let $H$ be a subgroup of index 2 in a group $G$. Show that if $a,b\in G$ and $a\notin H$ and $b\notin H$ then $ab\in H$. | |

Let $G$ be a group. Let $H$ be a subgroup of $G$ such that if $a,b\in G$ and $a\notin H$ and $b\notin H$ then $ab\in H$. Show that $H$ has index 2 in $G$. | |

Let $G$ be a group of order $841={\left(29\right)}^{2}$. Assume that $G$ is not cyclic. Show that if $g\in G$ then ${g}^{29}=1$. | |

Show that the subgroup $\left\{\right(1),(123),(132\left)\right\}$ of ${S}_{3}$ is a normal subgroup. | |

Show that the subgroup $\left\{\right(1),(12\left)\right\}$ of ${S}_{3}$ is not a normal subgroup. | |

Show that ${\mathrm{SL}}_{n}\left(\u2102\right)$ is a normal subgroup of ${\mathrm{GL}}_{n}\left(\u2102\right)$. | |

Let $G$ be a group. Show that $\left\{1\right\}$ and $G$ are normal subgroups of $G$. | |

Show that every subgroup of an abelian group is normal. | |

Write down the cosets in ${\mathrm{GL}}_{n}\left(\u2102\right)/{\mathrm{SL}}_{n}\left(\u2102\right)$ then show that $${\mathrm{GL}}_{n}\left(\u2102\right)/{\mathrm{SL}}_{n}\left(\u2102\right)\simeq {\mathrm{GL}}_{1}\left(\u2102\right).$$ | |

Show that the function $\mathrm{det}:{\mathrm{GL}}_{n}\left(\u2102\right)\to {\mathrm{GL}}_{1}\left(\u2102\right)$ given by taking the determinant of a matrix is a homomorphism. | |

Show that the function $f:{\mathrm{GL}}_{1}\left(\u2102\right)\to {\mathrm{GL}}_{1}\left(\mathbb{R}\right)$ given by $f\left(z\right)=\left|z\right|$ is a homomorphism. | |

Show that the determinant function $\mathrm{det}:{\mathrm{GL}}_{n}\left(\u2102\right)\to {\mathrm{GL}}_{1}\left(\u2102\right)$ is surjective and has kernel ${\mathrm{SL}}_{n}\left(\u2102\right)$. | |

Show that the homomorphism $f:{\mathrm{GL}}_{1}\left(\u2102\right)\to {\mathrm{GL}}_{1}\left(\mathbb{R}\right)$ given by $f\left(z\right)=\left|z\right|$ has image ${\mathbb{R}}_{>0}$ and kernel ${U}_{1}\left(\u2102\right)$ (the group of $1\times 1$ unitary matrices. Conclude that $${\mathrm{GL}}_{1}\left(\u2102\right)/{U}_{1}\left(\u2102\right)\simeq {\mathbb{R}}_{>0}.$$ | |

Show that the homomorphism $$\begin{array}{cccc}f:& \mathbb{R}& \to & {\mathrm{SO}}_{2}\left(\mathbb{R}\right)\\ & \theta & \to & \left(\begin{array}{cc}\mathrm{cos}\theta & \mathrm{sin}\theta \\ -\mathrm{sin}\theta & \mathrm{cos}\theta \end{array}\right)\end{array}$$ is surjective with kernel $2\pi \mathbb{Z}$. Conclude that $$\mathbb{R}/\left(2\pi \mathbb{Z}\right)\simeq {\mathrm{SO}}_{2}\left(\mathbb{R}\right).$$ | |

Show that the set of matrices $H=\left\{\left(\begin{array}{cc}a& b\\ 0& d\end{array}\right)\phantom{\rule{.5em}{0ex}}\right|\phantom{\rule{.5em}{0ex}}ad\ne 0\}$
is a subgroup of
| |

Let $G$ be a group and let $H$ be a subgroup of $G$. Show that $HH=H$. | |

Let $G$ be a group and let $K$ and $L$ be normal subgroups of $G$. Show that $K\cap L$ is a normal subgroup of $G$. | |

Let $G$ be a group and let $n$ be a positive integer. Assume that $H$ is the only subgroup of $G$ of order $n$. Show that $H$ is a normal subgroup of $G$. | |

Let $G$ be an abelian group and let $N$ be a normal subgroup of $G$. Show that $G/N$ is abelian. | |

Let $G$ be a cyclic group and let $N$ be a normal subgroup of $G$. Show that $G/N$ is cyclic. | |

Find surjective homomorphisms from $\mathbb{Z}/8\mathbb{Z}$ to $\mathbb{Z}/8\mathbb{Z}$, $\mathbb{Z}/4\mathbb{Z}$, $\mathbb{Z}/2\mathbb{Z}$, and $\left\{1\right\}$ (the group with one element). | |

Let $\mathbb{R}$ denote the group of real numbers with the operation of addition and let $\mathbb{Q}$ and $\mathbb{Z}$ be the subgroups of rational numbers and integers, respectively. Show that it is possible to regard $\mathbb{Q}/\mathbb{Z}$ as a subgroup of $\mathbb{R}/\mathbb{Z}$ and show that this subgroup consists exactly of the elements of finite order in $\mathbb{R}/\mathbb{Z}$. |

[GH]
J.R.J. Groves and
C.D. Hodgson,
*Notes for 620-297: Group Theory and Linear Algebra*, 2009.

[Ra]
A. Ram,
*Notes in abstract algebra*, University of Wisconsin, Madison 1994.