Math Problems Vault

Problem 6.1.1

(1)(2)(3)(4)

Let $G$ be the collection of $2\times 2$ real matrices with nonzero determinant. Define the product of two elements in $G$ as the usual matrix product.

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(1)

Show that $G$ is a group.

(2)

Find the center $Z$ of $G$ ; that is, the set of all elements $z$ of $G$ such that $az = za$ for all $a \in G$ .

(3)

Show that the set $O$ of real orthogonal matrices is a subgroup of $G$ (a matrix is orthogonal if $AA^{t} = I$ , where $A^{t}$ denotes the transpose of $A$ ). Show by example that $O$ is not a normal subgroup.

(4)

Find a nontrivial homomorphism from $G$ onto an abelian group.

Problem 6.1.2

(1)(2)

Let $G$ be the set of $3 \times 3$ real matrices with zeros below the diagonal and ones on the diagonal.

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(1)

Prove $G$ is a group under matrix multiplication.

(2)

Determine the center of $G$ .

Problem 6.1.3

For each of the following either give an example or else prove that no such example is possible.

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(1)

A nonabelian group.

(2)

A finite abelian group that is not cyclic.

(3)

An infinite group with a subgroup of index 5.

(4)

Two finite groups that have the same order but are not isomorphic.

(5)

A group $G$ with a subgroup $H$ that is not normal.

(6)

A nonabelian group with no normal subgroups except the whole group and the unit element.

(7)

A group $G$ with a normal subgroup $H$ such that the factor group $G/H$ is not isomorphic to any subgroup of $G$ .

(8)

A group $G$ with a subgroup $H$ which has index 2 but is not normal.

Problem 6.1.4

(1)(2)

Let $R$ be a ring with multiplicative identity 1. Call $x \in R$ a unit if $xy = yx = 1$ for some $y \in R$ . Let $G(R)$ denote the set of units.

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(1)

Prove $G(R)$ is a multiplicative group.

(2)

Let $R$ be the ring of complex numbers $a+bi$ , where $a$ and $b$ are integers. Prove $G(R)$ is isomorphic to $\mathbb {Z}_{4}$ (the additive group of integers modulo 4).

Problem 6.1.5

In the triangular network in $\mathbb {R}^{2}$ depicted below, the points $P_{0}, P_{1}, P_{2}$ , and $P_{3}$ are respectively $(0,0), (1,0), (0,1)$ , and $(1,1)$ . Describe the structure of the group of all Euclidean transformations of $\mathbb {R}^{2}$ which leave this network invariant.

{#fig-1 width="50%"}

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Problem 6.1.6

Does the set $G = \{ a \in \mathbb {R} \mid a > 0, a \neq 1\}$ form a group with the operation $a * b = a^{\log b}$ ?

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Problem 6.1.7

(1)(2)(3)

Let $G$ be a finite group. A conjugacy class is a set of the form

C(a) = \{ bab^{-1} \mid b \in G\}

for some $a \in G$ .

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(1)

Prove that the number of elements in a conjugacy class divides the order of $G$ .

(2)

Do all conjugacy classes have the same number of elements?

(3)

If $G$ has only two conjugacy classes, prove $G$ has order 2.

Problem 6.1.8

Let $G$ be a finite nontrivial group with the property that for any two elements $a$ and $b$ in $G$ different from the identity, there is an element $c$ in $G$ such that $b = c^{-1}ac$ . Prove that $G$ has order 2.

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Problem 6.1.9

Let $G$ be a finite group, with identity $e$ . Suppose that for every $a, b \in G$ distinct from $e$ , there is an automorphism $\sigma$ of $G$ such that $\sigma (a) = b$ . Prove that $G$ is abelian.

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Problem 6.1.10

For a $p$ -group of order $p^{4}$ , assume the center of $G$ has order $p^{2}$ . Determine the number of conjugacy classes of $G$ .

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Problem 6.1.11

In a commutative group $G$ , let the element $a$ have order $r$ , let $b$ have order $s$ ( $r, s < \infty$ ), and assume that the greatest common divisor of $r$ and $s$ is 1. Show that $ab$ has order $rs$ .

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Problem 6.1.12

(1)(2)(3)

Let $G$ be a group. For any subset $X$ of $G$ , define its centralizer $C(X)$ to be $\{ y \in G \mid xy = yx, \text{ for all } x \in X\}$ . Prove the following:

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(1)

If $X \subset Y$ , then $C(Y) \subset C(X)$ .

(2)

$X \subset C(C(X))$ .

(3)

$C(X) = C(C(C(X)))$ .

Problem 6.1.13

Let $D$ be a group of order $2n$ , where $n$ is odd, with a subgroup $H$ of order $n$ satisfying $xhx^{-1} = h^{-1}$ for all $h$ in $H$ and all $x$ in $D \setminus H$ . Prove that $H$ is commutative and that every element of $D \setminus H$ is of order 2.

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