# Tagged Questions

Use with the (group-theory) tag. A group is cyclic if it can be generated by a single element, $a$. Every element has the form $a^i$ for some $i\in\mathbb{Z}$, and so these groups are abelian.

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### $G$ a finite group with two non-trivial normal subgroups. $|G| = pq$. Why G cycle?

How can I prove that if $G$ is a finite group , and the order of $G$ is $pq$ while $p$ and $q$ are primes, and in addition , $G$ with two normalic subgroups , so --> $G$ is cycle? Ideas? Hwo can i ...
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### A group $G$ is locally cyclic if and only if $G$ is a union of a chain of cyclic subgroups ?

Is it true that a group $G$ is locally cyclic if and only if $G$ is a union of a chain of cyclic subgroups ?
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### Common generator of units in finite prime fields

It is well known that the unit group of a finite field is cyclic. What can we say about the generators? Specifically I am interested in the following question: Suppose we fix a positive integer $a$, ...
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### Write $\mathbb{Z}^3/L$ as a direct sum of cyclic groups

Let $L\subset \mathbb{Z}$ be the subgroup of $\mathbb{Z}^3$ generated by the elements $(-1,-1,4),(2,4,0),(3,3,8)$. Write $\mathbb{Z}^3/L$ as a direct sum of cyclic groups. I've tried creating a ...
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### Order of product of non-disjoint cycles

Let $a$ and $b$ be two non-disjoint cycles of order $m$ and $n$. Is there any general formula for the order of $a b$? I understand that we can convert any non-disjoint cycles into disjoint cycles and ...
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### cyclic groups homomorphism

I have the following task: "Determine the homomorphism between two cyclic groups. Which are injective, surjective or bijective?" I already found this for the cyclic group of integers: http://users....
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### Existence of subfields such that $[\mathbb{Q}(\zeta_{25}) : K_1]=4$ and $[\mathbb{Q}(\zeta_{25}) : K_2]=5$

I have the following two questions questions I am working on and am a little stuck : Let $L=\mathbb{Q}(\zeta_{25})$ where $\zeta_{25}$ is the primitive $n$-th root of unity. Prove that there are ...
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### Group of order $p^2$ [duplicate]

Let $G$ be a group of order $p^2$ where $p$ is a prime. I want to show that either $G$ is cyclic or is the product of two cyclic groups of order $p$. After some work, the problem reduces to showing ...
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### Prove that if $H \cong \mathbb Z$ or $H \cong \mathbb Z_n$ then $H = ⟨g⟩$ for some $g ∈ G$.

I'm studying for my final and I came across this homework problem that I had previously done but I don't remember how to do it anymore. It is as follows ($G, H$ are groups): Suppose that $H \le G$. ...
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### Element of infinite order and all generator of subgroups

Suppose that a has infinite order Find all generators of subgroup $\left \langle a^{3} \right \rangle$ Now, since a has infinite order then so does $\left ( a^{3} \right )^{n}$for if a has finite ...
Question: Let $G$ be a group and let $a \in G$. Prove that $\left \langle a^{-1} \right \rangle=\left \langle a \right \rangle$ Suppose $\left \langle a^{-1} \right \rangle$ so $\left \langle ... 1answer 26 views ### Proof for generator of the group of integer under addition modulo Theorem: An integer$k$in$\mathbb{Z}_{n}$is a generator of$\mathbb{Z}_{n}$If and Only if$gcd\left ( n,k \right )=1$My problem lies with proving the "If" condition and here is my attempt: ... 0answers 25 views ### Proof for generators of cyclic group [duplicate] Theorem: Let$G=\left \langle a \right \rangle $be a cyclic group of order n. Then$G = \left \langle a^{k} \right \rangle$if and only if$gcd\left ( n,k \right )=1$I've proven the "only if" ... 0answers 24 views ### Consider a set consisting of matrices and show it is a group. I am giving $$A=\{1,i,-1,-i\}$$ and$$B=\{\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}, \begin{pmatrix} 0 & -1 \\ 1 & \phantom{-}0\end{pmatrix}, \begin{pmatrix} -1 & \phantom{-}0 \\ ... 1answer 43 views ### Subgroups of the Semi-Direct Product$\mathbb{Z}/\mathbb{7Z} \rtimes (\mathbb{Z}/\mathbb{7Z})^{\times}$I want to list all the subgroups of the semi-direct product$\mathbb{Z}/\mathbb{7Z} \rtimes (\mathbb{Z}/\mathbb{7Z})^{x}$, under the homomorphism$\theta: (\mathbb{Z}/\mathbb{7Z})^{\times} \rightarrow ...
Let $K$ be a ﬁeld containing a primitive $n$th root of unity and let $F = K(t)$ be the ﬁeld of rational functions over $K$. I'm having trouble proving that for each $n > 1$ the ﬁeld $F$ is Galois ...