Question

Why the number of elements of $ Z$ of order $d$ is $\varphi (d)$?

score 46 · Answer 1 · 2023-10-12 00:32:41

46

Views

Why the number of elements of $ Z$ of order $d$ is $\varphi (d)$?

Published on 12 Oct 2023 - 0:32

score 68 · Answer 2 · 2026-03-27 02:37:27

68

Views

Proving that $A_4/V_4$ can be generated by $\sigma V_4$ where $\sigma$ is a 3-cycle

Published on 27 Mar 2026 - 2:37

score 738 · Answer 3 · 2023-10-18 19:32:21

738

Views

Characterizations of divisors of the order of an element.

Published on 18 Oct 2023 - 19:32

score 94 · Answer 4 · 2026-03-29 14:03:00

94

Views

Sylow $p$-subgroups are all cyclic then $G$ has normal subgroup $N$ and such that $G/N$ and $N$ are both cyclic

Published on 29 Mar 2026 - 14:03

score 34 · Answer 5 · 2023-10-23 02:10:27

34

Views

If $ord(a)=n$ and $a=b^k$ then $\langle a\rangle=\langle b\rangle$ iff $gcd(k,n)=1$.

Published on 23 Oct 2023 - 2:10

score 37 · Answer 6 · 2023-10-23 03:49:55

37

Views

$n$-cycles and repeated elements

Published on 23 Oct 2023 - 3:49

score 39 · Answer 7 · 2026-03-29 22:44:18

39

Views

Are all the virtual cyclic subgroups of $PSL(2,\mathbb{R})$ elementary?

Published on 29 Mar 2026 - 22:44

score 103 · Answer 8 · 2026-03-26 18:58:50

103

Views

Infinite non cyclic group with proper subgroups being cyclic

Published on 26 Mar 2026 - 18:58

score 191 · Answer 9 · 2023-11-01 17:55:37

191

Views

Proof by contradiction using properties of normal subgroup: A group is cyclic if $x^m=e$ has at most $m$ solutions.

Published on 01 Nov 2023 - 17:55

score 133 · Answer 10 · 2026-03-27 01:00:07

133

Views

Let $G_1, G_2$ be non-trivial groups. If $G_1 \times G_2$ is cyclic, then $G_1 \times G_2$ is finite.

Published on 27 Mar 2026 - 1:00

score 106 · Answer 11 · 2023-11-03 22:32:01

106

Views

Let $G=\left\langle a \right\rangle$ Show that any subgroup of G is also cyclic

Published on 03 Nov 2023 - 22:32

score 252 · Answer 12 · 2023-11-09 05:29:29

252

Views

Show that a subgroup of order $3$ is normal in a group of order $15$

Published on 09 Nov 2023 - 5:29

score 122 · Answer 13 · 2023-11-19 21:54:14

122

Views

What is the intuition for recognizing cyclic groups?

Published on 19 Nov 2023 - 21:54

score 59 · Answer 14 · 2023-11-21 09:18:56

59

Views

$a$ is a torsion element of the group $G$, $yay^{-1}=a^k \neq a$. Can we conclude that $\langle a,y\rangle $ is a finite group?

Published on 21 Nov 2023 - 9:18

score 65 · Answer 15 · 2023-11-23 20:43:58

65

Views

To prove that two cyclic subgroups H and K within a cyclic group G are the same when their orders are equal

Published on 23 Nov 2023 - 20:43

List Question

Why the number of elements of $ Z$ of order $d$ is $\varphi (d)$?

Proving that $A_4/V_4$ can be generated by $\sigma V_4$ where $\sigma$ is a 3-cycle

Characterizations of divisors of the order of an element.

Sylow $p$-subgroups are all cyclic then $G$ has normal subgroup $N$ and such that $G/N$ and $N$ are both cyclic

If $ord(a)=n$ and $a=b^k$ then $\langle a\rangle=\langle b\rangle$ iff $gcd(k,n)=1$.

$n$-cycles and repeated elements

Are all the virtual cyclic subgroups of $PSL(2,\mathbb{R})$ elementary?

Infinite non cyclic group with proper subgroups being cyclic

Proof by contradiction using properties of normal subgroup: A group is cyclic if $x^m=e$ has at most $m$ solutions.

Let $G_1, G_2$ be non-trivial groups. If $G_1 \times G_2$ is cyclic, then $G_1 \times G_2$ is finite.

Let $G=\left\langle a \right\rangle$ Show that any subgroup of G is also cyclic

Show that a subgroup of order $3$ is normal in a group of order $15$

What is the intuition for recognizing cyclic groups?

$a$ is a torsion element of the group $G$, $yay^{-1}=a^k \neq a$. Can we conclude that $\langle a,y\rangle $ is a finite group?

To prove that two cyclic subgroups H and K within a cyclic group G are the same when their orders are equal

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