Question

Let $G$ be an abelian group of order $pq$ where $(p,q)=1$. Assume that $a,b\in G$ whose orders are $p$ and $q$, respectively. Show that $G$ is cyclic.

score 131 · Answer 1 · 2023-05-04 16:44:39

131

Views

Let $G$ be an abelian group of order $pq$ where $(p,q)=1$. Assume that $a,b\in G$ whose orders are $p$ and $q$, respectively. Show that $G$ is cyclic.

Published on 04 May 2023 - 16:44

score 81 · Answer 2 · 2026-02-23 02:13:49

81

Views

If $G/G’ $ is cyclic, prove by induction that $G/Z(G)$ is cyclic if $G$ is $p$ group.

Published on 23 Feb 2026 - 2:13

score 330 · Answer 3 · 2023-05-17 08:43:36

330

Views

Proving that groups are cyclic and finding generators.

Published on 17 May 2023 - 8:43

score 65 · Answer 4 · 2023-05-18 17:20:42

65

Views

Compositum of Galois extension

Published on 18 May 2023 - 17:20

score 86 · Answer 5 · 2026-03-28 01:48:51

86

Views

If G is a finite cyclic group of size n is generated by g, is it generated by $g^2$?

Published on 28 Mar 2026 - 1:48

score 138 · Answer 6 · 2026-03-26 17:27:57

138

Views

2-ended groups are virtually cyclic: reference

Published on 26 Mar 2026 - 17:27

score 44 · Answer 7 · 2023-05-29 02:26:22

44

Views

Cyclic Groups of Order n

Published on 29 May 2023 - 2:26

score 168 · Answer 8 · 2023-06-03 10:29:50

168

Views

If $G$ has exactly $28$ elements of order $5$, how many subgroups of $G$ has order $5$?

Published on 03 Jun 2023 - 10:29

score 69 · Answer 9 · 2023-06-13 19:48:48

69

Views

Find all isomorphisms between the group of rotations of the square and the group (Z4, +4)

Published on 13 Jun 2023 - 19:48

score 69 · Answer 10 · 2023-06-22 06:44:55

69

Views

Regular representation of $Z_3$

Published on 22 Jun 2023 - 6:44

score 152 · Answer 11 · 2026-03-28 08:09:25

152

Views

Prove irreducible cubic polynomial over $\mathbb{Q}$ with a cyclic galois group has real roots

Published on 28 Mar 2026 - 8:09

score 68 · Answer 12 · 2026-03-26 17:30:31

68

Views

Finitely generated group with a unique maximal subgroup. Is it cyclic?

Published on 26 Mar 2026 - 17:30

score 103 · Answer 13 · 2026-03-26 22:57:56

103

Views

Writing a quotient group as a product of cyclic groups

Published on 26 Mar 2026 - 22:57

score 168 · Answer 14 · 2026-03-30 21:38:21

168

Views

Branching Rules of Symmetric group to cyclic group formula

Published on 30 Mar 2026 - 21:38

score 198 · Answer 15 · 2026-03-30 03:21:31

198

Views

A cyclic group of order 25 doesn't have an automorphism of order 3.

Published on 30 Mar 2026 - 3:21

List Question

Let $G$ be an abelian group of order $pq$ where $(p,q)=1$. Assume that $a,b\in G$ whose orders are $p$ and $q$, respectively. Show that $G$ is cyclic.

If $G/G’ $ is cyclic, prove by induction that $G/Z(G)$ is cyclic if $G$ is $p$ group.

Proving that groups are cyclic and finding generators.

Compositum of Galois extension

If G is a finite cyclic group of size n is generated by g, is it generated by $g^2$?

2-ended groups are virtually cyclic: reference

Cyclic Groups of Order n

If $G$ has exactly $28$ elements of order $5$, how many subgroups of $G$ has order $5$?

Find all isomorphisms between the group of rotations of the square and the group (Z4, +4)

Regular representation of $Z_3$

Prove irreducible cubic polynomial over $\mathbb{Q}$ with a cyclic galois group has real roots

Finitely generated group with a unique maximal subgroup. Is it cyclic?

Writing a quotient group as a product of cyclic groups

Branching Rules of Symmetric group to cyclic group formula

A cyclic group of order 25 doesn't have an automorphism of order 3.

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