Math Problems Vault

Problem 1.2.1

Let $G$ be the group of real $2 \times 2$ matrices, of determinant one. Describe the set of conjugacy classes of elements of $G$ .

?

Problem 1.2.2

Let $G = GL_n(F_q)$ , the group of invertible $n \times n$ matrices over the finite field $F_q$ with $q = p^r$ , $p$ a prime, $U = U_n(F_q)$ , the subgroup of upper triangular matrices with $1$ 's on the diagonal.

?

(a)

Calculate the orders of $G$ and $U$ . Deduce that $U$ is a Sylow $p$ -subgroup of $G$ .

(b)

Deduce that every $p$ -subgroup of $G$ is conjugate to a subgroup of $U$ .

(c)

Determine the number of $G$ -conjugates of $U$ .

(d)

Show that $g \in G$ has $p$ -power order iff $g = I + N$ with $N^n = 0$ .

(e)

Show that $G$ contains an element of order $q^n - 1$ .

Problem 1.2.3

(a)(b)

?

(a)

Let $G$ be a finite group and $H$ a subgroup. Prove that if $G = \bigcup_{g \in G} gHg^{-1}$ then $H = G$ .

(b)

Recall that a subgroup $M$ of a group $G$ is said to be maximal if the only subgroups $H$ satisfying $M \subseteq H \subseteq G$ are $H = M$ and $H = G$ . Let $G$ be a finite group with the property that all of its maximal subgroups are conjugate. Prove that $G$ is cyclic of prime power order.

Problem 1.2.4

(a)(b)

Suppose that $H$ and $K$ are normal subgroups of a finite group $G$ and that $G/H \simeq K$ .

?

(a)

Give an example to show that $G/K$ need not be isomorphic to $H$ .

(b)

Show that if $H$ is simple, then $G/K \simeq H$ .

Problem 1.2.5

Prove that if $G$ is a finitely generated infinite group then for each positive integer $n$ , $G$ has only finitely many subgroups of index $n$ .

?

Problem 1.2.6

G = \left\{ \begin{pmatrix} a & b \\ 0 & a^{-1} \end{pmatrix} \, \middle |\, a \in \mathbb {R}^*, b \in \mathbb {R}\right\} \subseteq GL(2, \mathbb {R}).

Show that $G$ is solvable but not nilpotent.

?

Problem 1.2.7

(a)(b)

Let $p$ be a prime and let $V$ be an $n$ -dimensional vector space over $F_p$ . Let $G = GL_n(V)$ . Recall that $\left|G\right| = (p^n - 1)(p^n - p) \cdots (p^n - p^{n-1})$ .

?

(a)

Recall that a linear transformation is called semisimple if its minimal polynomial is separable. Prove that a transformation $T \in G$ is semisimple if and only if $T^{p^m - 1} = I$ for some positive integer $m$ .

(b)

Let $H$ be a subgroup of $G$ of order a power of $p$ . Show that $H$ can be simultaneously upper triangularized, that is, there is a basis of $V$ with respect to which all of the elements of $H$ are upper triangular.

Problem 1.2.8

(a)(b)(c)

Let $p$ and $q$ be distinct prime numbers. Let $G$ be a group of order $p^3 \cdot q$ such that its commutator subgroup $K$ is of order $q$ . Let $H$ be a $p$ -Sylow subgroup of $G$ .

?

(a)

Show that $H$ is abelian and $G = HK$ .

(b)

Show that there are elements $h \in H$ and $k \in K$ such that $hk \neq kh$ .

(c)

From (b) show that $p$ divides $q - 1$ .

Problem 1.2.9

(a)(b)

?

(a)

Let $p$ , $q$ be primes, $p > q > 2$ . Let $G$ be a group of order $pq^2$ . Show $G$ has a subgroup of order $pq$ .

(b)

What can you say if $q = 2$ (and $p > q$ )?

Problem 1.2.10

Let $A$ be the abelian group on generators $e$ , $f$ , and $g$ , subject to the relations

9e + 3f + 6g = 0,

3e + 3f = 0,

3e - 3f + 6g = 0.

Give a decomposition of $A$ as a direct sum of cyclic groups of prime order or infinite order.

?

Problem 1.2.11

(a)(b)

Let $M$ be an $n \times n$ matrix of integers. Suppose that $M$ is invertible when viewed as a matrix of rational numbers, i.e., that there exists an $n \times n$ matrix $N$ with rational entries so that $MN$ equals the $n \times n$ identity matrix. View $M$ as an endomorphism of $\mathbb {Z}^n$ .

?

(a)

Show that $\mathbb {Z}^n/M\mathbb {Z}^n$ is finite.

(b)

Show that the order of $\mathbb {Z}^n/M\mathbb {Z}^n$ is equal to the absolute value of the determinant of $M$ .

Problem 1.2.12

Let $D = \mathbb {Z}[\xi ]$ with $\xi = \frac{-1+\sqrt{-3}}{2}$ . Calculate the order of the additive group $G = D^2/K$ where $K$ is the $D$ -submodule of $D^2$ generated by $(2\xi +1, \xi -1)$ , $(\xi +2, \xi -4)$ and $(21, 21)$ . Then express $G$ as a product of cyclic groups.

?

Problem 1.2.13

Prove that $SL(2, \mathbb {Z})$ is generated by $\begin{pmatrix} 1 & \pm 1 \\ 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ where $SL(2, \mathbb {Z})$ is the group of $2 \times 2$ matrices with integral coefficients and determinant $= 1$ .

?

Problem 1.2.14

Let $G$ be a finite group, $K$ a normal subgroup of $G$ and $P$ a Sylow subgroup of $K$ . If $N$ is the normalizer of $P$ in $G$ (i.e., $N = \left\{ g \in G \mid gPg^{-1} = P\right\}$ ) then show that $G = K \cdot N$ .

?

Problem 1.2.15

Let $P$ be a Sylow subgroup of a finite group $G$ . Let $N = \left\{ x \in G \mid xPx^{-1} = P\right\}$ . Let $H$ be a subgroup of $G$ , $H \supseteq N$ . Prove: If $y \in G$ such that $yHy^{-1} = P$ , then $y \in H$ .

?

Problem 1.2.16

(a)(b)

Let $G$ be a finite group. The probability of $G$ to be commutative is defined by

P(G) = \frac{\left|\left\{ (a,b) \in G \times G \mid ab = ba\right\} \right|}{\left|G \times G\right|}.

?

(a)

If $G$ is not commutative, prove that $P(G) \leq 5/8$ .

(b)

Give an example of $G$ such that $P(G)$ is exactly $5/8$ .