# Questions tagged [cyclic-groups]

Use with the (group-theory) tag. A group is cyclic if it can be generated by a single element. That is to say, every element in a cyclic group can be written as some specified element to a power.

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### Transforming a permutation into a rotation

Let $n\geq 3$ and $G=C_{n}=\{1,r,...,r^{n-1}\}$ be the cyclic group of $n$ elements where $r$ is the rotation of $360/n$ degrees. Here, let us consider a vector $x\in\mathbb{R}^{n}$ as consisting of ...
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### Quotient C8 by C2 group

I attempt to divide a C8 by C2. My reasoning: C2 is a normal subgroup of C8. C2 forms 2 cosets: {0,2,4,6} and 1+{0,2,4,6}. C8/C2 isomorphic to C4. But I know that C2*C4 is an Abelian group. It is ...
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### Automorphism of commutative groups.

For every group G there is a natural group homomorphism G → Aut(G) whose image is the group Inn(G) of inner automorphisms and whose kernel is the center of G. Thus, if G has trivial center it can ...
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### Cardinality of the kernel of multiplication map

Let $m$ and $n$ be positive integers and let $F : \mathbb{Z}_n \to \mathbb{Z}_n$ be given by $F(\overline{x}) = m \overline{x} = \overline{mx}$. I am struggling to show the fact that $\ker F$ has ...
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### Working out subgroups of cyclic groups [closed]

If I have cyclic groups: $X = C_3 = \langle x\rangle$ and $Y = C_2 =\langle y \rangle$ , with $x$ being of order 3 and $y$ being of order 2. How would I work out the subgroups of $X \times Y$ And if ...
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### Groups of units of cyclic groups of order $2^n$ [duplicate]

In Dummit & Foote's Abstract Algebra (2nd ed.), Corollary 20(2) states the following: $(\mathbb{Z}/2^n\mathbb{Z})^\times$ is the direct product of a cyclic group of order 2 and a cyclic group ...
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### Inverse of modulo sum of elements from a cyclical multiplicative group without descrete log

So the question: Given a hash function $$h(x) = \sum_{i=1}^n g^{x_i} \mod p$$ where p is a prime number and g is a generator of the multiplicative cyclic group $\mathbb{Z}_p^×$. n is the length of the ...
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### Prove that $M_{16} / Z(M_{16}) \cong C_2 \times C_2$

I am trying to answer the following question: Prove that $M_{16} / Z(M_{16}) \cong C_2 \times C_2$. Where $M_{16}=\langle x, y | x^{8}=1, y^{2}=1, yx=x^{5}y\rangle$ is the modular group of order ...
### all possible combinations of $a_j$s for $\sum_{j=1}^na_j\epsilon^{j}=0$ where $\epsilon=e^{\frac{2\pi i}{n}}$
Let $n\geq 3$ and $d$ be a divisor of $n$. Let us consider the set $\{1,\epsilon,\dots,\epsilon^{n-1}\}$ where $\epsilon=e^{\frac{2\pi i}{n}}$. Can we classify all possible cases for $d$ distinct ...