Question

Finding all generators of subgroups of order $8$ in $\Bbb{Z}_{32}$.

score 1.5k · Answer 1 · 2018-08-13 01:47:55

1.5k

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Finding all generators of subgroups of order $8$ in $\Bbb{Z}_{32}$.

Published on 13 Aug 2018 - 1:47

score 2.9k · Answer 2 · 2018-08-13 03:51:08

2.9k

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Show that $y=x^{k}$ with $gcd(k,n)=1$ is a generator of $G$.

Published on 13 Aug 2018 - 3:51

score 1.1k · Answer 3 · 2018-08-13 05:06:42

1.1k

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Let $p$ be prime and let $r$ be a positive integer. How many generators does $\mathbb{Z}_{p^{r}}$ have?

Published on 13 Aug 2018 - 5:06

score 2.7k · Answer 4 · 2018-08-16 23:08:41

2.7k

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Prove that the generators of $\mathbb{Z}_n$ are the integers $r$ such that $0\leq r <n$ and $\gcd(r,n)=1$.

Published on 16 Aug 2018 - 23:08

score 288 · Answer 5 · 2018-08-17 01:10:33

288

Views

Cyclic notation for residue classes mod $n$

Published on 17 Aug 2018 - 1:10

score 484 · Answer 6 · 2018-08-17 21:57:16

484

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Inclusions between cyclic subgroups of a given cyclic group

Published on 17 Aug 2018 - 21:57

score 306 · Answer 7 · 2018-08-18 05:54:43

306

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Let $z=\operatorname{cis}\theta \in \mathbb{T}$ with $\theta \in \mathbb{Q}$. Show that the order of $z$ is infinite.

Published on 18 Aug 2018 - 5:54

score 4.2k · Answer 8 · 2026-03-25 20:11:56

4.2k

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Let $G$ be an abelian group of order $pq$ with $\gcd(p,q)=1$. Show that $G$ is cyclic.

Published on 25 Mar 2026 - 20:11

score 173 · Answer 9 · 2026-03-25 23:39:25

173

Views

Number of (cyclic) subgroups of $\mathbb{Z}_{16}\times\mathbb{Z}_{30}$

Published on 25 Mar 2026 - 23:39

score 262 · Answer 10 · 2026-03-25 18:56:18

262

Views

Order of each element in a symmetry group.

Published on 25 Mar 2026 - 18:56

score 243 · Answer 11 · 2026-03-26 06:29:09

243

Views

$n$-th root of unity implies $d$-th primitive root of unity for some $d\big|_n$

Published on 26 Mar 2026 - 6:29

score 442 · Answer 12 · 2026-03-25 18:54:38

442

Views

Amount of elements of order $p^2$

Published on 25 Mar 2026 - 18:54

score 181 · Answer 13 · 2026-03-25 12:53:51

181

Views

How many epimorphisms from $F_2$ to $\mathbb{Z}_5$?

Published on 25 Mar 2026 - 12:53

score 52 · Answer 14 · 2018-08-26 13:32:50

52

Views

How to minimize the cardinality of the intersection of two sets of pairs from a cyclic group?

Published on 26 Aug 2018 - 13:32

score 206 · Answer 15 · 2026-03-26 07:30:47

206

Views

Extension of a cyclic group by an abelian group

Published on 26 Mar 2026 - 7:30

List Question

Finding all generators of subgroups of order $8$ in $\Bbb{Z}_{32}$.

Show that $y=x^{k}$ with $gcd(k,n)=1$ is a generator of $G$.

Let $p$ be prime and let $r$ be a positive integer. How many generators does $\mathbb{Z}_{p^{r}}$ have?

Prove that the generators of $\mathbb{Z}_n$ are the integers $r$ such that $0\leq r <n$ and $\gcd(r,n)=1$.

Cyclic notation for residue classes mod $n$

Inclusions between cyclic subgroups of a given cyclic group

Let $z=\operatorname{cis}\theta \in \mathbb{T}$ with $\theta \in \mathbb{Q}$. Show that the order of $z$ is infinite.

Let $G$ be an abelian group of order $pq$ with $\gcd(p,q)=1$. Show that $G$ is cyclic.

Number of (cyclic) subgroups of $\mathbb{Z}_{16}\times\mathbb{Z}_{30}$

Order of each element in a symmetry group.

$n$-th root of unity implies $d$-th primitive root of unity for some $d\big|_n$

Amount of elements of order $p^2$

How many epimorphisms from $F_2$ to $\mathbb{Z}_5$?

How to minimize the cardinality of the intersection of two sets of pairs from a cyclic group?

Extension of a cyclic group by an abelian group

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