Question

Show that $(\mathbb{R}^*, \cdot)$ is not cyclic.

score 429 · Answer 1 · 2021-08-23 23:45:23

429

Views

Show that $(\mathbb{R}^*, \cdot)$ is not cyclic.

Published on 23 Aug 2021 - 23:45

score 152 · Answer 2 · 2021-08-26 11:19:41

152

Views

Are $\mathbb{Z} \times \mathbb{Z}_2$ and $ \mathbb{Z}$ isomorphic?

Published on 26 Aug 2021 - 11:19

score 94 · Answer 3 · 2021-08-31 02:51:30

94

Views

Can any subgroup of a cyclic group $\left<a\right>$ be presented as $\{x\in \left<a\right>|x^m=e\}$?

Published on 31 Aug 2021 - 2:51

score 400 · Answer 4 · 2026-03-28 02:03:27

400

Views

Which groups have non-trivial cyclic quotient groups?

Published on 28 Mar 2026 - 2:03

score 296 · Answer 5 · 2026-03-26 16:06:58

296

Views

Quotient groups of the Gaussian integers of the form $\mathbb{Z}[i]/ c\mathbb{Z}[i]$

Published on 26 Mar 2026 - 16:06

score 114 · Answer 6 · 2026-04-01 07:51:46

114

Views

If $(h,k) = 1$, then $h^{\phi(k)}\equiv 1(\textrm{mod}\ k)$?

Published on 01 Apr 2026 - 7:51

score 764 · Answer 7 · 2021-09-04 16:00:33

764

Views

If $G$ is a finite cyclic group of prime power order then given any two subgroups of $G$, one of the subgroups contains the other.

Published on 04 Sep 2021 - 16:00

score 92 · Answer 8 · 2021-09-04 18:48:05

92

Views

Show that the set $H= \{[0]_{10}, [5]_{10}\}$ is a subgroup of $\Bbb Z_{10}$.

Published on 04 Sep 2021 - 18:48

score 90 · Answer 9 · 2026-04-04 06:17:58

90

Views

Image of Abelian/Cyclic Tower is an Abelian/Cyclic Tower

Published on 04 Apr 2026 - 6:17

score 84 · Answer 10 · 2021-09-07 04:11:59

84

Views

$(\mathbb{Z}/p^{a}\mathbb{Z})^{\times}$ is cyclic group of order $p^{a-1}(p-1)$

Published on 07 Sep 2021 - 4:11

score 24 · Answer 11 · 2026-03-29 14:25:08

24

Views

Partition $nk$ objects up to $\frac{\binom{nk}{m}}{n\binom{k}{m}}$ times, each time making $k$-subsets, so that no combination of $m$ is ever repeated

Published on 29 Mar 2026 - 14:25

score 37 · Answer 12 · 2026-03-30 16:45:23

37

Views

On a solution to finding four groups of order $20$ not isomorphic to each other.

Published on 30 Mar 2026 - 16:45

score 119 · Answer 13 · 2026-03-28 09:57:05

119

Views

An abelian $p$-group has a bounded basic subgroup iff it is the direct sum of a divisible group and bounded group

Published on 28 Mar 2026 - 9:57

score 213 · Answer 14 · 2021-09-13 00:24:21

213

Views

Hungerford exercise V.2.8

Published on 13 Sep 2021 - 0:24

score 435 · Answer 15 · 2021-09-13 10:40:50

435

Views

Let $G=\langle a\rangle $ cyclic group of order $9$ If $H=\langle a^2\rangle $ prove that $H=G$

Published on 13 Sep 2021 - 10:40

List Question

Show that $(\mathbb{R}^*, \cdot)$ is not cyclic.

Are $\mathbb{Z} \times \mathbb{Z}_2$ and $ \mathbb{Z}$ isomorphic?

Can any subgroup of a cyclic group $\left<a\right>$ be presented as $\{x\in \left<a\right>|x^m=e\}$?

Which groups have non-trivial cyclic quotient groups?

Quotient groups of the Gaussian integers of the form $\mathbb{Z}[i]/ c\mathbb{Z}[i]$

If $(h,k) = 1$, then $h^{\phi(k)}\equiv 1(\textrm{mod}\ k)$?

If $G$ is a finite cyclic group of prime power order then given any two subgroups of $G$, one of the subgroups contains the other.

Show that the set $H= \{[0]_{10}, [5]_{10}\}$ is a subgroup of $\Bbb Z_{10}$.

Image of Abelian/Cyclic Tower is an Abelian/Cyclic Tower

$(\mathbb{Z}/p^{a}\mathbb{Z})^{\times}$ is cyclic group of order $p^{a-1}(p-1)$

Partition $nk$ objects up to $\frac{\binom{nk}{m}}{n\binom{k}{m}}$ times, each time making $k$-subsets, so that no combination of $m$ is ever repeated

On a solution to finding four groups of order $20$ not isomorphic to each other.

An abelian $p$-group has a bounded basic subgroup iff it is the direct sum of a divisible group and bounded group

Hungerford exercise V.2.8

Let $G=\langle a\rangle $ cyclic group of order $9$ If $H=\langle a^2\rangle $ prove that $H=G$

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