Question

Prove that if $G=\langle x\rangle$ for every $x\in G\setminus \{1_G\}$, then $|G|$ is prime

score 54 · Answer 1 · 2021-03-28 14:14:11

54

Views

Prove that if $G=\langle x\rangle$ for every $x\in G\setminus \{1_G\}$, then $|G|$ is prime

Published on 28 Mar 2021 - 14:14

score 86 · Answer 2 · 2026-03-28 09:50:21

86

Views

Hungerford's notation of $(|G|,|H|)=1$.

Published on 28 Mar 2026 - 9:50

score 86 · Answer 3 · 2026-03-26 06:09:15

86

Views

Proving that a group is either cyclic or not-simple

Published on 26 Mar 2026 - 6:09

score 103 · Answer 4 · 2021-03-30 08:20:17

103

Views

A notational question about Cyclic Groups

Published on 30 Mar 2021 - 8:20

score 139 · Answer 5 · 2021-03-30 08:47:35

139

Views

All cyclic groups of order $n$ are isomorphic

Published on 30 Mar 2021 - 8:47

score 128 · Answer 6 · 2021-04-05 05:51:01

128

Views

For $|G|=pq$ ($p$ & $q$ are distinct primes), $H$ and $K$ are normal subgroups of order $p$ and $q$ resp., then, prove that $G$ is cyclic

Published on 05 Apr 2021 - 5:51

score 714 · Answer 7 · 2026-04-03 22:31:05

714

Views

Let $\varphi:G\to G'$ be a surjective homomorphism. Prove that if $G$ is cyclic, then $G'$ is cyclic, and if $G$ is abelian, then $G'$ is abelian.

Published on 03 Apr 2026 - 22:31

score 110 · Answer 8 · 2026-03-30 03:01:50

110

Views

F.G. abelian group so that every quotient is cyclic

Published on 30 Mar 2026 - 3:01

score 247 · Answer 9 · 2026-03-26 02:56:09

247

Views

If every proper subgroup of $G$ is cyclic and normal then is $G$ finite cyclic?

Published on 26 Mar 2026 - 2:56

score 132 · Answer 10 · 2026-02-23 08:05:41

132

Views

Find a product of cyclic groups isomorphic to Z/20

Published on 23 Feb 2026 - 8:05

score 355 · Answer 11 · 2026-03-25 13:55:15

355

Views

If a finite group has exactly $2$ maximal subgroups, then it is $3$-cyclic

Published on 25 Mar 2026 - 13:55

score 29 · Answer 12 · 2026-03-28 21:06:36

29

Views

Why is this quotient group not cyclic?

Published on 28 Mar 2026 - 21:06

score 408 · Answer 13 · 2026-03-28 11:31:59

408

Views

Why is $C_4 \times C_4$ not isomorphic to $C_4 \times C_2 \times C_2$?

Published on 28 Mar 2026 - 11:31

score 206 · Answer 14 · 2026-02-23 15:33:07

206

Views

Definition of minus power in cyclic group

Published on 23 Feb 2026 - 15:33

score 70 · Answer 15 · 2021-05-03 03:11:01

70

Views

Number of elements of order $p$ in product of $\mathbb{Z}_{p^a} \times \mathbb{Z}_{q^b} \times \mathbb{Z}_{r^{c}}$

Published on 03 May 2021 - 3:11

List Question

Prove that if $G=\langle x\rangle$ for every $x\in G\setminus \{1_G\}$, then $|G|$ is prime

Hungerford's notation of $(|G|,|H|)=1$.

Proving that a group is either cyclic or not-simple

A notational question about Cyclic Groups

All cyclic groups of order $n$ are isomorphic

For $|G|=pq$ ($p$ & $q$ are distinct primes), $H$ and $K$ are normal subgroups of order $p$ and $q$ resp., then, prove that $G$ is cyclic

Let $\varphi:G\to G'$ be a surjective homomorphism. Prove that if $G$ is cyclic, then $G'$ is cyclic, and if $G$ is abelian, then $G'$ is abelian.

F.G. abelian group so that every quotient is cyclic

If every proper subgroup of $G$ is cyclic and normal then is $G$ finite cyclic?

Find a product of cyclic groups isomorphic to Z/20

If a finite group has exactly $2$ maximal subgroups, then it is $3$-cyclic

Why is this quotient group not cyclic?

Why is $C_4 \times C_4$ not isomorphic to $C_4 \times C_2 \times C_2$?

Definition of minus power in cyclic group

Number of elements of order $p$ in product of $\mathbb{Z}_{p^a} \times \mathbb{Z}_{q^b} \times \mathbb{Z}_{r^{c}}$

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