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Abstract Algebra

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Subgroup of a Group: Subgroup Generated by Elements, Definition

ABSALG-ZYNSDX

Let $G$ be a group, and $S$ be a non-empty subset of $G$.

The subgroup $\langle S\rangle$ of $G$ generated by $S$ is defined as: the intersection of all subgroups of $G$ containing $S$, and $S$ is called a generating set for $\langle S\rangle$.

The elements of $S$ are called generators of $\langle S\rangle$.

Denote:

by $({\mathbb Z}/m{\mathbb Z})^\ast$, $m\ge 2$, the group of reduced (coprime to $m$) residue classes mod $m$.

by ${\mathbb Z}$ the additive group of integers.

by ${\mathbb Q}^\ast$ the multiplicative group of non-zero rational numbers.

Which of the following are true?

Select ALL that apply.

A

There is a unique $g\in({\mathbb Z}/5{\mathbb Z})^\ast$ with $({\mathbb Z}/5{\mathbb Z})^\ast=\langle g\rangle$

B

There are exactly two distinct $g\in{\mathbb Z}$ such that ${\mathbb Z}=\langle g\rangle$

C

We have ${\mathbb Q}^\ast= \langle {\mathcal P}\rangle$, where ${\mathcal P}$ is the set of prime numbers $p\ge 2$

D

Let $r,s$ be coprime integers, then ${\mathbb Z}=\langle r,s\rangle$

E

For a fixed $m\ge 2$, there are only finitely many pairs of integers $r, s$ such that $m{\mathbb Z}=\langle r, s\rangle$