Question

Can a non-cyclic group be isomorphic to a cyclic group and vice-versa?

score 164 · Answer 1 · 2022-11-09 04:20:55

164

Views

Can a non-cyclic group be isomorphic to a cyclic group and vice-versa?

Published on 09 Nov 2022 - 4:20

score 131 · Answer 2 · 2026-04-01 16:30:13

131

Views

If $p<q$, $p\nmid q-1$, then “every group of order $pq$ is cyclic” iff “there is no nontrivial homomorphism from $ℤ_p$ into $\mathrm{Aut}(ℤ_q)$”.

Published on 01 Apr 2026 - 16:30

score 53 · Answer 3 · 2026-02-24 03:53:37

53

Views

$\mathbb{Z}/n\mathbb{Z}$ as the union of $\frac{n}{e}(\mathbb{Z}/e\mathbb{Z})^\times$

Published on 24 Feb 2026 - 3:53

score 43 · Answer 4 · 2022-11-25 04:18:20

43

Views

Proof that different direct product lead to elements of different order.

Published on 25 Nov 2022 - 4:18

score 107 · Answer 5 · 2026-04-06 02:43:15

107

Views

If a discrete group acts on a compact metric space with an infinite invariant set, is it cyclic?

Published on 06 Apr 2026 - 2:43

score 68 · Answer 6 · 2022-12-03 05:42:59

68

Views

some property and its solution for a cyclic-symmetric inequality

Published on 03 Dec 2022 - 5:42

score 66 · Answer 7 · 2022-12-14 15:52:10

66

Views

If $G = \left<x\right>$ is a cyclic group of order $n= p^ka$ with $(p,a)=1$, can we write $x=cb$ with $c$ of order $p^k$ and $b$ of order $a$?

Published on 14 Dec 2022 - 15:52

score 74 · Answer 8 · 2026-03-30 08:32:56

74

Views

Amalgamated product of finite cyclic groups is not abelian

Published on 30 Mar 2026 - 8:32

score 129 · Answer 9 · 2022-12-18 17:08:37

129

Views

Non-trivial intersection of two cyclic groups

Published on 18 Dec 2022 - 17:08

score 267 · Answer 10 · 2022-12-19 22:08:14

267

Views

Suppose that $G$ is a group of order $105$ with the property that $G$ has exactly one subgroup for each divisor of $105$. Prove that $G$ is cyclic

Published on 19 Dec 2022 - 22:08

score 170 · Answer 11 · 2022-12-26 12:39:20

170

Views

Are cyclic groups defined for only two operations (multiplication, addition)?

Published on 26 Dec 2022 - 12:39

score 88 · Answer 12 · 2022-12-28 23:38:47

88

Views

For two cyclic groups $\phi:C_4 \to C_3$ with $x^i \mapsto y^i$, we have $\phi(x^4)=\phi(x^0)=e$ but also $\phi(x^4) = y^4=y$. Where am I going wrong?

Published on 28 Dec 2022 - 23:38

score 42 · Answer 13 · 2026-03-29 22:08:25

42

Views

Quotient of ideals of prime power in a localized integer ring

Published on 29 Mar 2026 - 22:08

score 195 · Answer 14 · 2026-03-30 08:32:37

195

Views

Prove that $f(x) = -1-3x \pmod{2^e}$ cycles with period $2^e$

Published on 30 Mar 2026 - 8:32

score 83 · Answer 15 · 2026-03-30 01:46:12

83

Views

Technicality about cycle of a permutation, for coding purposes

Published on 30 Mar 2026 - 1:46

List Question

Can a non-cyclic group be isomorphic to a cyclic group and vice-versa?

If $p<q$, $p\nmid q-1$, then “every group of order $pq$ is cyclic” iff “there is no nontrivial homomorphism from $ℤ_p$ into $\mathrm{Aut}(ℤ_q)$”.

$\mathbb{Z}/n\mathbb{Z}$ as the union of $\frac{n}{e}(\mathbb{Z}/e\mathbb{Z})^\times$

Proof that different direct product lead to elements of different order.

If a discrete group acts on a compact metric space with an infinite invariant set, is it cyclic?

some property and its solution for a cyclic-symmetric inequality

If $G = \left<x\right>$ is a cyclic group of order $n= p^ka$ with $(p,a)=1$, can we write $x=cb$ with $c$ of order $p^k$ and $b$ of order $a$?

Amalgamated product of finite cyclic groups is not abelian

Non-trivial intersection of two cyclic groups

Suppose that $G$ is a group of order $105$ with the property that $G$ has exactly one subgroup for each divisor of $105$. Prove that $G$ is cyclic

Are cyclic groups defined for only two operations (multiplication, addition)?

For two cyclic groups $\phi:C_4 \to C_3$ with $x^i \mapsto y^i$, we have $\phi(x^4)=\phi(x^0)=e$ but also $\phi(x^4) = y^4=y$. Where am I going wrong?

Quotient of ideals of prime power in a localized integer ring

Prove that $f(x) = -1-3x \pmod{2^e}$ cycles with period $2^e$

Technicality about cycle of a permutation, for coding purposes

Trending Questions

Popular # Hahtags

Popular Questions