Question

Finite groups where $x^2 = e$ has order $2^n$

score 1.2k · Answer 1 · 2015-09-23 03:30:53

1.2k

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Finite groups where $x^2 = e$ has order $2^n$

Published on 23 Sep 2015 - 3:30

score 2.1k · Answer 2 · 2015-09-23 15:43:48

2.1k

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Multiplicative group of an infinite field is not cyclic

Published on 23 Sep 2015 - 15:43

score 8.1k · Answer 3 · 2015-09-25 04:26:06

8.1k

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If $G$ be a cyclic group of prime order $p$, prove that every non-identity element of $G$ is a generator of $G$.

Published on 25 Sep 2015 - 4:26

score 1.3k · Answer 4 · 2026-04-01 17:54:05

1.3k

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Is the symmetric group $S_4$ cyclic

Published on 01 Apr 2026 - 17:54

score 414 · Answer 5 · 2015-09-25 07:04:47

414

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Without using Cauchy's theorem: If $G$ an abelian group of order $10$ contains an element of order $5$, show that $G$ must be a cyclic group.

Published on 25 Sep 2015 - 7:04

score 2k · Answer 6 · 2015-09-25 16:48:18

2k

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Let a, b be fixed positive integers and $H=\{ax+by|x,y\in \Bbb Z\}.$ Show that $H$ is a cyclic group with $\gcd(a,b)$ as a generator.

Published on 25 Sep 2015 - 16:48

score 2.9k · Answer 7 · 2015-09-26 04:46:59

2.9k

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Let $H$ be a group. Let $a, b$ be fixed positive integers and $H=\{ax+by\mid x,y\in \Bbb Z\}.$ Show that $d\mathbb Z =H$ where $d=\gcd(a,b)$.

Published on 26 Sep 2015 - 4:46

score 2.6k · Answer 8 · 2015-09-26 09:24:18

2.6k

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Let $G$ be a cyclic group of order $n$. Prove that every subgroup $H$ of $G$ is of the form $<a^m>$ where $m$ is a divisor of $n$.

Published on 26 Sep 2015 - 9:24

score 3k · Answer 9 · 2015-09-29 02:10:04

3k

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Let G = <x> be a cyclic group of order 24. List all the elements in G that are of order 4.

Published on 29 Sep 2015 - 2:10

score 196 · Answer 10 · 2015-09-30 13:09:49

196

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Virtually cyclic groups: Is the finite permutation group virtually cyclic? & Uncountable groups cannot be virtually cyclic.

Published on 30 Sep 2015 - 13:09

score 194 · Answer 11 · 2015-10-01 12:37:02

194

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g is a generator for G, is also g^n a generator?

Published on 01 Oct 2015 - 12:37

score 190 · Answer 12 · 2015-10-01 18:47:04

190

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Proposition 2.3 (Cyclic Groups) in Dummit and Foote

Published on 01 Oct 2015 - 18:47

score 217 · Answer 13 · 2015-10-02 04:39:21

217

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Prove that group $G=\{n_1r_1+n_2r_2 | n_1,n_2\in\mathbb{Z}\}$ is cyclic

Published on 02 Oct 2015 - 4:39

score 3.1k · Answer 14 · 2015-10-02 05:18:09

3.1k

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Using Lagrange's theorem, prove that a non-abelian group of order $10$ must have a subgroup of order $5$.

Published on 02 Oct 2015 - 5:18

score 1.4k · Answer 15 · 2015-10-02 06:27:24

1.4k

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Let $G$ an commutative group of order $2n$ where $n$ is odd prime. Suppose $G$ has an element of order $n$. Show that $G$ is cyclic.

Published on 02 Oct 2015 - 6:27

List Question

Finite groups where $x^2 = e$ has order $2^n$

Multiplicative group of an infinite field is not cyclic

If $G$ be a cyclic group of prime order $p$, prove that every non-identity element of $G$ is a generator of $G$.

Is the symmetric group $S_4$ cyclic

Without using Cauchy's theorem: If $G$ an abelian group of order $10$ contains an element of order $5$, show that $G$ must be a cyclic group.

Let a, b be fixed positive integers and $H=\{ax+by|x,y\in \Bbb Z\}.$ Show that $H$ is a cyclic group with $\gcd(a,b)$ as a generator.

Let $H$ be a group. Let $a, b$ be fixed positive integers and $H=\{ax+by\mid x,y\in \Bbb Z\}.$ Show that $d\mathbb Z =H$ where $d=\gcd(a,b)$.

Let $G$ be a cyclic group of order $n$. Prove that every subgroup $H$ of $G$ is of the form $<a^m>$ where $m$ is a divisor of $n$.

Let G = <x> be a cyclic group of order 24. List all the elements in G that are of order 4.

Virtually cyclic groups: Is the finite permutation group virtually cyclic? & Uncountable groups cannot be virtually cyclic.

g is a generator for G, is also g^n a generator?

Proposition 2.3 (Cyclic Groups) in Dummit and Foote

Prove that group $G=\{n_1r_1+n_2r_2 | n_1,n_2\in\mathbb{Z}\}$ is cyclic

Using Lagrange's theorem, prove that a non-abelian group of order $10$ must have a subgroup of order $5$.

Let $G$ an commutative group of order $2n$ where $n$ is odd prime. Suppose $G$ has an element of order $n$. Show that $G$ is cyclic.

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