Question

Prove that the total number of subgroups of a finite cyclic group $G=\langle a \rangle$ such that $o(G)=n$ is the number of divisors of $n$.

score 1.8k · Answer 1 · 2015-10-02 14:43:31

1.8k

Views

Prove that the total number of subgroups of a finite cyclic group $G=\langle a \rangle$ such that $o(G)=n$ is the number of divisors of $n$.

Published on 02 Oct 2015 - 14:43

score 2.8k · Answer 2 · 2015-10-03 06:18:00

2.8k

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Show by Lagrange's theorem that a group $G$ of order $27$ should have a subgroup of order $3$.

Published on 03 Oct 2015 - 6:18

score 63 · Answer 3 · 2015-10-05 23:18:51

63

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Does $H \unlhd G$ imply $G/H$ is cyclic and therefore abelian?

Published on 05 Oct 2015 - 23:18

score 86 · Answer 4 · 2015-10-06 03:51:29

86

Views

Subgroup of $\mathbb{Z}_a \times \mathbb{Z}_b$ related to GCD and LCM of $a$ and $b$

Published on 06 Oct 2015 - 3:51

score 368 · Answer 5 · 2015-10-06 15:35:20

368

Views

Usage go the Fundamental Theorem of Cyclic Groups

Published on 06 Oct 2015 - 15:35

score 25 · Answer 6 · 2015-10-06 21:19:26

25

Views

Does $k$ generate $K$?

Published on 06 Oct 2015 - 21:19

score 78 · Answer 7 · 2015-10-06 22:58:49

78

Views

Proving $\gcd(n,\lvert G\rvert) = 1$

Published on 06 Oct 2015 - 22:58

score 152 · Answer 8 · 2015-10-10 06:08:11

152

Views

What is an example of a cyclic group with exactly 6 generators?

Published on 10 Oct 2015 - 6:08

score 92 · Answer 9 · 2026-03-29 22:09:19

92

Views

Simultaneously diagonalize the regular representation of C2 (+) C2 (+) C2.

Published on 29 Mar 2026 - 22:09

score 1.2k · Answer 10 · 2026-03-30 12:45:20

1.2k

Views

Prove that $ a^6 = b^{11} $

Published on 30 Mar 2026 - 12:45

score 4.6k · Answer 11 · 2015-10-14 02:15:19

4.6k

Views

Prove that the four-group $\{1,a,b,c \}$ is not cyclic.

Published on 14 Oct 2015 - 2:15

score 173 · Answer 12 · 2015-10-14 08:35:52

173

Views

Let $ G $ be a group and $ N \lhd G $. Why is $G $ supersoluble if $ N $ is cyclic and $ G/N $ is supersoluble?

Published on 14 Oct 2015 - 8:35

score 1k · Answer 13 · 2015-10-14 23:07:32

1k

Views

What is a cyclic group with exactly one generator?

Published on 14 Oct 2015 - 23:07

score 614 · Answer 14 · 2015-10-17 11:10:10

614

Views

The order of elements and cyclic subgroups of $\mathbb Z/18\mathbb Z$

Published on 17 Oct 2015 - 11:10

score 130 · Answer 15 · 2015-10-18 19:35:23

130

Views

Proving that the multiplicative group modulo $2^r$ are cyclic iff $r<3$

Published on 18 Oct 2015 - 19:35

List Question

Prove that the total number of subgroups of a finite cyclic group $G=\langle a \rangle$ such that $o(G)=n$ is the number of divisors of $n$.

Show by Lagrange's theorem that a group $G$ of order $27$ should have a subgroup of order $3$.

Does $H \unlhd G$ imply $G/H$ is cyclic and therefore abelian?

Subgroup of $\mathbb{Z}_a \times \mathbb{Z}_b$ related to GCD and LCM of $a$ and $b$

Usage go the Fundamental Theorem of Cyclic Groups

Does $k$ generate $K$?

Proving $\gcd(n,\lvert G\rvert) = 1$

What is an example of a cyclic group with exactly 6 generators?

Simultaneously diagonalize the regular representation of C2 (+) C2 (+) C2.

Prove that $ a^6 = b^{11} $

Prove that the four-group $\{1,a,b,c \}$ is not cyclic.

Let $ G $ be a group and $ N \lhd G $. Why is $G $ supersoluble if $ N $ is cyclic and $ G/N $ is supersoluble?

What is a cyclic group with exactly one generator?

The order of elements and cyclic subgroups of $\mathbb Z/18\mathbb Z$

Proving that the multiplicative group modulo $2^r$ are cyclic iff $r<3$

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