Question

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

score 1.5k · Answer 1 · 2026-04-14 18:05:36

1.5k

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

Published on 14 Apr 2026 - 18:05

score 2.9k · Answer 2 · 2026-04-12 23:10:23

2.9k

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

Published on 12 Apr 2026 - 23:10

score 1.1k · Answer 3 · 2026-04-13 12:24:42

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 Apr 2026 - 12:24

score 2.7k · Answer 4 · 2026-04-15 12:35:49

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 15 Apr 2026 - 12:35

score 291 · Answer 5 · 2026-04-15 17:30:49

291

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Cyclic notation for residue classes mod $n$

Published on 15 Apr 2026 - 17:30

score 487 · Answer 6 · 2026-04-15 03:09:09

487

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Inclusions between cyclic subgroups of a given cyclic group

Published on 15 Apr 2026 - 3:09

score 309 · Answer 7 · 2026-04-15 19:19:11

309

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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 15 Apr 2026 - 19:19

score 4.2k · Answer 8 · 2026-04-17 02:29:28

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 17 Apr 2026 - 2:29

score 176 · Answer 9 · 2026-05-10 15:52:07

176

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

Published on 10 May 2026 - 15:52

score 264 · Answer 10 · 2026-04-12 12:13:56

264

Views

Order of each element in a symmetry group.

Published on 12 Apr 2026 - 12:13

score 246 · Answer 11 · 2026-05-10 23:27:52

246

Views

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

Published on 10 May 2026 - 23:27

score 452 · Answer 12 · 2026-05-10 12:33:40

452

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Amount of elements of order $p^2$

Published on 10 May 2026 - 12:33

score 186 · Answer 13 · 2026-05-10 12:33:53

186

Views

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

Published on 10 May 2026 - 12:33

score 55 · Answer 14 · 2026-04-14 03:30:31

55

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

Published on 14 Apr 2026 - 3:30

score 209 · Answer 15 · 2026-05-10 14:56:59

209

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Extension of a cyclic group by an abelian group

Published on 10 May 2026 - 14:56

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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