This site is currently being migrated at a new site. Please read the information below.

LaTeX

Unicode

Showing posts with label Group Theory. Show all posts
Showing posts with label Group Theory. Show all posts

Sunday, November 20, 2016

The group is abelian

Let $\mathcal{G}$ be a finite group such that $\left ( \left | \mathcal{G} \right |  , 3 \right ) =1$. If for the elements $a, \beta \in \mathcal{G}$ holds that

$$\left ( a \beta \right )^3 = a^3 \beta^3$$

then prove that $\mathcal{G}$ is abelian.

Solution

Tuesday, April 12, 2016

Cyclic group

Let $\mathcal{G}$ be a group of order $n$. Prove that if $\gcd(\varphi(n), n )=1$ then $\mathcal{G}$ is cyclic, where $\varphi$ denotes Euler's totient function.

Solution

Sunday, April 10, 2016

Normal subgroup

Let $\mathcal{S}_4$ denote the symmetric $4$ - permutation group. Prove that

$$\mathcal{H}=\big\{{{\rm{id}},\,({1\,2})\,({3\,4}),\,({1\,3})\,({2\,4}),\,({1\,4})\,({2\,3})}\big\}$$

is a normal subgroup of $\mathcal{S}_4$.

Solution

Thursday, April 7, 2016

They are isomorphic

Prove that:

$$\mathbb{Z}_2 \times \mathbb{Z}_4 \times \mathbb{Z}_8 \bigg/\langle [1]_4, [2]_4, [4]_8 \rangle \cong \mathbb{Z}_4 \times \mathbb{Z}_8$$

Solution

Sunday, April 3, 2016

Order of an element

Let  $G$ be a group and $a\,,b\in G$, such that  $a^5=e$ and $a\,b\,a^{-1}=b^2$, where $e$ is the identity element of $G$. Find the order of $b$.

Solution

Thursday, March 24, 2016

An abelian group

Let $G$ be a group such that for (some) three consecutive integer numbers $i$ and for all $a, b \in G$ , it holds that:

$$(ab)^i=a^i b^i$$

Prove that $G$ is abelian.

Solution

Friday, March 11, 2016

Cosets and subgroup

Let $G$ be a group and $H$ be a subgroup of $G$. Let $b \in H$. Prove that:

$$Hb =H$$

where $Hb$ is the left coset of $H$.

Solution

Tuesday, February 23, 2016

Being a group implies that $n$ is prime

Prove that $(\mathbb{Z}_n^*, \otimes)$ is a group if and only if $n$ is prime.

Solution

Finite cyclic group and generator

Let $G=\langle a \rangle$ be a finite cyclic group of order $n$ with generator $a$. Prove that $a^k$ generates $G$ if and only if $\gcd(n, k) =1$.

Solution