Math Problems Vault

Problem 6.3.1

(1)(2)

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(1)

Prove that every finitely generated subgroup of $\mathbb {Q}$ , the additive group of rational numbers, is cyclic.

(2)

Does the same conclusion hold for finitely generated subgroups of $\mathbb {Q}/\mathbb {Z}$ , where $\mathbb {Z}$ is the group of integers?

Note: See also Problems 6.6.3 and 6.7.2.

Problem 6.3.2

Let $G$ be the group $\mathbb {Q}/\mathbb {Z}$ . Show that for every positive integer $t$ , $G$ has a unique cyclic subgroup of order $t$ .

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Problem 6.3.3

Show that a group $G$ is isomorphic to a subgroup of the additive group of the rationals if and only if $G$ is countable and every finite subset of $G$ is contained in an infinite cyclic subgroup of $G$ .

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Problem 6.3.4

(1)(2)

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(1)

Let $G$ be a cyclic group, and let $a, b \in G$ be elements which are not squares. Prove that $ab$ is a square.

(2)

Give an example to show that this result is false if the group is not cyclic.

Problem 6.3.5

Prove that any group of order 77 is cyclic.

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Problem 6.3.6

Let $G$ be a group of order $2p$ , where $p$ is an odd prime. Assume that $G$ has a normal subgroup of order 2. Prove that $G$ is cyclic.

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Problem 6.3.7

A finite abelian group $G$ has the property that for each positive integer $n$ the set $\{ x \in G \mid x^{n} = 1\}$ has at most $n$ elements. Prove that $G$ is cyclic, and deduce that every finite field has cyclic multiplicative group.

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Problem 6.3.8

Let $G$ be a finite group of order $n$ with the property that for each divisor $d$ of $n$ there is at most one subgroup in $G$ of order $d$ . Show $G$ is cyclic.

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