Math Problems Vault

Problem 6.2.1

How many homomorphisms are there from the group $\mathbb {Z}_{2} \times \mathbb {Z}_{2}$ to the symmetric group on three objects?

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Problem 6.2.2

(1)(2)

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(1)

Let $G$ be a finite group and let $X$ be the set of pairs of commuting elements of $G$ :

X = \{ (g,h) \in G \times G : gh = hg\} .

Prove that $\left|X\right| = c\left|G\right|$ where $c$ is the number of conjugacy classes in $G$ .

(2)

Compute the number of pairs of commuting permutations on five letters.

Problem 6.2.3

The set of $5 \times 5$ complex matrices $A$ satisfying $A^{3} = A^{2}$ is a union of conjugacy classes. How many conjugacy classes?

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Problem 6.2.4

Let $\mathbb {C}^{*}$ be the multiplicative group of nonzero complex numbers. Suppose that $H$ is a subgroup of finite index of $\mathbb {C}^{*}$ . Prove that $H = \mathbb {C}^{*}$ .

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Problem 6.2.5

(1)(2)

Let $G$ be a finite group and $H \subset G$ a subgroup.

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(1)

Show that the number of subgroups of $G$ of the form $xHx^{-1}$ for some $x \in G$ is $\leq$ the index of $H$ in $G$ .

(2)

Prove that some element of $G$ is not in any subgroup of the form $xHx^{-1}$ , $x \in G$ .

Problem 6.2.6

Prove that the group of automorphisms of a cyclic group of prime order $p$ is cyclic and find its order.

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Problem 6.2.7

(1)(2)

Let $G$ be a finite group, and let $\varphi$ be an automorphism of $G$ which leaves fixed only the identity element of $G$ .

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(1)

Show that every element of $G$ may be written in the form $g^{-1}\varphi (g)$ .

(2)

If $\varphi$ has order 2 (i.e., $\varphi \cdot \varphi = \operatorname {id}$ ) show that $\varphi$ is given by the formula $g \mapsto g^{-1}$ and that $G$ is an abelian group whose order is odd.

Problem 6.2.8

Prove that every finite group of order $> 2$ has a nontrivial automorphism.

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Problem 6.2.9

Let $A$ be an additively written abelian group, and $u, v : A \to A$ homomorphisms. Define the group homomorphisms $f, g : A \to A$ by

f(a) = a - v(u(a)), \qquad g(a) = a - u(v(a)) \qquad (a \in A).

Prove that the kernel of $f$ is isomorphic to the kernel of $g$ .

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Problem 6.2.10

Let $G$ be an additive group, and $u, v : G \to G$ homomorphisms. Show that the map $f : G \to G$ , $f(x) = x - v(u(x))$ is surjective if the map $h : G \to G$ , $h(x) = x - u(v(x))$ is surjective.

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Problem 6.2.11

Let $H$ be the group of integers mod $p$ , under addition, where $p$ is a prime number. Suppose that $n$ is an integer satisfying $1 \leq n \leq p$ , and let $G$ be the group $H \times H \times \cdots \times H$ ( $n$ factors). Show that $G$ has no automorphism of order $p^{2}$ .

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Problem 6.2.12

(1)(2)

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(1)

Suppose that $H_{1}$ and $H_{2}$ are subgroups of a group $G$ such that $H_{1} \cup H_{2}$ is a subgroup of $G$ . Prove that either $H_{1} \subset H_{2}$ or $H_{2} \subset H_{1}$ .

(2)

Show that for each integer $n \geq 3$ , there exists a group $G$ with subgroups $H_{1}, H_{2}, \ldots , H_{n}$ , such that no $H_{i}$ is contained in any other, and such that $H_{1} \cup H_{2} \cup \ldots \cup H_{n}$ is a subgroup of $G$ .

Problem 6.2.13

(1)(2)

Let $G$ be a group and $H$ a subgroup of index $n < \infty$ . Prove or disprove the following statements:

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(1)

If $a \in G$ , then $a^{n} \in H$ .

(2)

If $a \in G$ , then for some $k$ , $1 \leq k \leq n$ , we have $a^{k} \in H$ .

Problem 6.2.14

Find all automorphisms of the additive group of rational numbers.

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Problem 6.2.15

Let $\alpha$ and $\beta$ be real numbers such that the subgroup $\Gamma$ of $\mathbb {R}$ generated by $\alpha$ and $\beta$ is closed. Prove that $\alpha$ and $\beta$ are linearly dependent over $\mathbb {Q}$ .

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Problem 6.2.16

Prove that the group $G = \mathbb {Q}/\mathbb {Z}$ has no proper subgroup of finite index.

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Problem 6.2.17

Let $A$ be the group of rational numbers under addition, and let $M$ be the group of positive rational numbers under multiplication. Determine all homomorphisms $\varphi : A \to M$ .

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Problem 6.2.18

Suppose that $H_{1}$ and $H_{2}$ are distinct subgroups of a group $G$ such that $[G:H_{1}] = [G:H_{2}] = 3$ . What are the possible values of $[G:H_{1} \cap H_{2}]$ ?

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Problem 6.2.19

Let $G$ be a group and $H$ and $K$ subgroups such that $H$ has a finite index in $G$ . Prove that $K \cap H$ has a finite index in $K$ .

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Problem 6.2.20

Suppose the group $G$ has a nontrivial subgroup $H$ which is contained in every nontrivial subgroup of $G$ . Prove that $H$ is contained in the center of $G$ .

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Problem 6.2.21

Show that for each positive integer $k$ there exists a positive integer $N$ such that there are at least $k$ nonisomorphic groups of order $N$ .

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Problem 6.2.22

Let $G$ be a group generated by $n$ elements. Find an upper bound $N(n,k)$ for the number of subgroups $H$ of $G$ with the index $[G:H] = k$ .

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