Articles of cyclic groups

How to find the quotient group $Z_{1023}^*/\langle 2\rangle$?

When $m=1023$, what are the quotient groups below? $$Z_m^*/ \langle 2\rangle$$ $Z_m^*=\{1,2,4,5, \dots \}$ $\langle 2\rangle=\{1,2,4,8,16,32,64,128,256,512\}$ $$\begin{align*} Z_m^*/\langle 2\rangle &=\{1,2,4,8,16,32,64,128,256,512\},\\ {} & \mathrel{\hphantom{=}}\{2,3,5,9,17,33,65,129,257,513\}, &&\text{(added by 1)},\\ {} & \mathrel{\hphantom{=}}\{3,4,6,10,18,34,66,130,258,514\}, && \text{(added by 2)},\\ {} & \mathrel{\hphantom{=}}\{5,6,8,12,20,36,68,132,260,516\}, &&\text{(added by 4)}\\ & \,\, \vdots \end{align*}$$ Are the answers $\langle 2 \rangle$ incremented by all the elements i$\in […]

Prove that G is a cyclic group

Suppose that $|G| = pq$ where $p$ and $q$ are primes such that $p < q$ and $p$ does not divide $q − 1$. Prove that $G$ is a cyclic group. A cyclic group is a group that has a unique generator element, so is the way to go with this to find that element? […]

How many subgroups of $\Bbb{Z}_4 \times \Bbb{Z}_6$?

This question already has an answer here: Determining the number of subgroups of $\Bbb Z_{14} \oplus \Bbb Z_{6}$ 1 answer

Prove or disprove that $\exists a,b,c\in\mathbb{Z}_+\ \forall n\in\mathbb{Z}_+\ \exists k\in\mathbb{Z}_+\colon\ a^k+b^k+c^k = 0(\mathrm{mod}\ 2^n)$

Prove or disprove that there exist a triplet of positive integers $(a,b,c)$ with $\mathrm{gcd}(a,b,c)=1$ such that for any positive integer $n$ there exist such positive integer $k$ that $$ a^k+b^k+c^k = 0(\mathrm{mod}\ 2^n) $$ First thing that I’ve noticed is that one and only one of the numbers $a,b,c$ is even, otherwise either all of […]

Is ${\mathbb Z} \times {\mathbb Z}$ cyclic?

Not sure where to go with this, but I don’t think it is cyclic..

give an example of a cyclic group with 6 generators.

Give an example of a cyclic group with 6 generators. Give the generators, explain how you know that these are generators and that they are the only generators. I don’t even know how to begin this problem. The textbook we’re using is not the easiest to understand. All we have covered in class is order, […]

Determining whether two groups are isomorphic

I am reading “A First Course in Algebra”, and there, I am trying to solve the exercises, but there is something i don’t understand. How do we understand whether two groups are isomorphic or not? For example, there is an example that asks whether the given groups are isomorphic: $\Bbb Z_8 \times \Bbb Z_{10} \times […]

Find a generator of the multiplicative group of $\mathbb{Z}/23\mathbb{Z}$ as a cyclic group

I need to find a generator of the multiplicative group of $\mathbb{Z}/23\mathbb{Z}$ as a cyclic group. Since $\mathbb{Z}/23\mathbb{Z}$ only has $23$ elements and ord$(x)$ where $x$ is a generator must divide $23$, then does this mean the generator can only be $1$ or $23$? Or have I got the wrong idea? It would help if […]

Help me to prove that group is cyclic

Prove that a group of order 5 must be cyclic, and every Abelian group of order 6 will also be cyclic. Let G be the group of order 5. To prove group of order 5 is cyclic do we have prove it by every element $(\langle a\rangle =\langle e,a,a^2,a^3,a^4,a^5=e\rangle)\forall a \in G$

Generators of a cyclic group

In a paper there is a lemma: Let $G= \langle a,b \rangle$ be a finite cyclic group. Then $G=\langle ab^n \rangle$ for some integer $n$. The proof is omitted because it’s “straightforward” but I’m not able to proof it. How does this work?