# Tagged Questions

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### Cyclic subgroups of semidirect products

Let $H$ and $N$ be two finite groups and $\phi:H\to Aut(N)$ a homomorphism. Let $G=N\rtimes_\phi H$ and let $\pi:G\to G/N=H$ be the quotient map. Let $p$ be a prime which does not divide $|N|$ but ...
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### Another Presentation of Certain Cyclic Groups

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^n\rangle$$ is cyclic of order $3(n+1)$, for $n=0 \mod 3$ or $n= 1 \mod 3$, $n\ge 0$. This ...
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### Cyclic Group Presentation

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^7\rangle$$ is cyclic of order 24. This presentation was obtained using the Todd-Coxeter ...
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### What are all the automorphisms of a group of order $9$ generated by two elements?

Let $G$ be a group of order $9$ generated by two elements $a$ and $b$ such that $a^3 = b^3 =e$. How to determine all possible automorphisms of $G$?
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### A question on cyclic group

I have no trouble proving the following statement. Let $G$ be a cyclic group of order $n$ and let $k$ be an integer relatively prime to $n$. Prove the map $x \mapsto x^k$ is surjective. It is clear by ...
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### What is this notation? Cyclic group $\mathbb{Z}^*_8$

$\mathbb{Z}^*_8$ As I understand it - $\mathbb{Z}_8$ is the group of integers under addition modulo 8. So am I correct in thinking its elements are: $\{0,1,2,3,4,5,6,7\}$? I thought the $*$ meant ...
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### if $f$ is of prime order, then can the orbit of $s$ under $f$ have one element?

if $f\in A(S)$ has order $p$, $p$ a prime, show that for every $s\in S$ the orbit of $s$ under $f$ has one or $p$ elements.* Since the cyclic group generated by $f$ has $p$ elements, therefore ...
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### How to count generators for a cyclic group

Show that there are $\varphi (n)$ generators for a cyclic group $G$ of order $n$. Give their form explicitly. Here $\varphi (n)$ is the Euler's function. I don't know what to do, please help.
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### Abstract Algebra: prove it is cyclic

I have question in referring to below link. Question. Suppose if I have [M:K]=2 and I know that K is subset of M. M:=$\mathbb{Z}_2[x]/(f(x))$ where f(x)=x$^4$+x+1. Then how this will be cyclic? I ...
### How many subgroups of $\Bbb{Z}_4 \times \Bbb{Z}_6$?
I have been trying to calculate the number of subgroups of the direct cross product $\Bbb{Z}_4 \times \Bbb{Z}_6.$ Using Goursat's Theorem, I can calculate 16. Here's the info: Goursat's Theorem: Let ...