New answers tagged cyclic-groups
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No cyclic subgroups of orders 8 and 9 in $G/H$
"Since $8=|g'+H|||g'|$, then $|g'| = 8n$", how do you get this?
If such a cyclic subgroup exsits, assume the generator is $g'+H$, then we have $(g'+H)^8=H$ and $(g'+H)^k\neq H$, for $k=1,2,.....
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group $G$ such that $|G| = p^k$ for some prime $p$ and some integer $k > 1$, where $|g| = p$ for every $g \in G \setminus {\epsilon}$
More generally:
$$G=C_p\times C_p$$
where $C_p$ is the cyclic group of order $p$ (in multiplicative notation), has those properties: it has order $p^2$ and every nontrivial element has order $p$. Your ...
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Is there a group theoretic proof that $(\mathbf Z/(p))^\times$ is cyclic?
The fact in the OP ($x^r\equiv 1\pmod p$ has at most $r$ solutions) shows immediately that $({\bf Z}/(p))^\times$ can't have any subgroup isomorphic to $C_q\times C_q$, for any prime $q$ dividing $p-1$...
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How do I find the order of elements in $\mathbb{C}^*$ (the multiplicative complex group)?
Our question is, given an element $z\in \mathbb{C}^*$, what is the lowest natural number $n$ satisfying $z^n=1$?
Since we require that $z^n=1$, it only makes sense that $z$ is a root of unity; i.e. $z=...
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About every subgroup of $ ( \mathbb{Z} , + ) $ being cyclic.
Suppose $H\neq \{0\}=\langle 0\rangle$. Then $\exists h\in H$ such that $h\neq 0$. Let $$m=\min \{g\in H\mid 0\lt g\leq |h|\}$$ where $|h|=h$ if $h\gt 0$, and $|h|=-h$ if $h\lt 0$. Note $m$ exists by ...
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Axioms for finite cyclic groups
If you were to take this approach, in the context of CS / cryptography education as you mention in your comment, what I think you'd want would be something like:
A cyclic group of order $n$ is a set $...
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Prove every Sylow Subgroup of $G$ is Cyclic
Since, $G$ has unique subgroup of each order then every sylow-$P$ subgroup of G is unique.
Let, $P$ is not cyclic. Let, $h$ be the element of maximal order in $P$. Consider, $H = <h>$ since, G ...
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Prove every Sylow Subgroup of $G$ is Cyclic
Take $g\in P$ of maximal order, say $p^i$. If $i<k$ then there is some element $h\in P\setminus\langle g\rangle$. It has order $p^j$, and by maximality of $i$ we must have $j\leq i$. Then $\langle ...
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Axioms for finite cyclic groups
You wrote
Diffie-Hellman key exchange protocol or ElGamal encryption scheme rely on finite cyclic groups. For computer science students who have no idea of what a group is, it may be an overkill to ...
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Axioms for finite cyclic groups
You could take Peano's axioms and replace the axiom which states $0$ is not the successor of any natural number with its negation (number 2 below).
Your axioms then are, $G$ is a set together with a ...
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Why isn’t every subgroup of a Finite Group Cyclic?
Take a non-cyclic group $G$ and any group $H$. Then, $K:=G\times\{1_H\}\cong G$ and $K\le G\times H$.
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Why isn’t every subgroup of a Finite Group Cyclic?
You have shown that every cyclic subgroup $\langle g\rangle$ of $G$ is cyclic. This is not very surprising. It doesn't imply that all subgroups of $G$ are cyclic, though. The easiest counterexample is ...
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If $a^k\equiv a\mod p$ for all $a$, show $(p - 1)\mid (k-1)$
Well...Since, for non-zero $a$, we have $a^{p-1}\equiv a^{k-1}\equiv 1\pmod p$ we can deduce that $a^d\equiv 1 \pmod p$ for $d=\gcd(p-1,k-1)$. Here, of course, $1≤d≤p-1$.
But you know that $\mathbb ...
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What is the connection between the notions of cyclic permutation and cyclic group?
You asked about the connection between a cyclic permutation and a cyclic group. Cayley's Theorem
states that every group is isomorphic to a subgroup
of the symmetric group using the regular ...
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Group ring over a cyclic group
The hint in the comments is a good one that works for cyclic groups, but there is actually one that works simply for all finite groups (or, for that matter, any group with a nontrivial finite ...
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Let $G = (\mathbb Z, +)$ and $H,K ≤ G$ where $H = ⟨12⟩$ and K =$\mathbb⟨18⟩$; define $H\mathbb\cap K$ and $H + K$.
While @Shaun is right that you should generally ask only one question at a time, I think these could be construed as related enough to be one post. Your questions seem to mostly be about what the ...
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Looking for a simple proof that groups of order $2p$ are up to isomorphism $\Bbb{Z}_{2p}$ and $D_p$ for prime $p>2$.
For $p=2$ our group has 4 elements, so it is either cyclic or the Klein-4-group. So let $p > 2$.
By Cauchy's theorem you know there is a cyclic subgroup of order $p$ and $2$, call this $H \leq G$. ...
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Show that if $G$ is a cyclic group of order $n$ and $d\mid n$, then $G$ has a subgroup of order $d$.
Nevermind I answered it myself. Here is the answer in case someone in the future has the same problem as me and can't figure it out.
Consider the element $a^k$ in $G$. Notice that $(a^k)^d = a^{(kd)} =...
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Which cyclic groups are automorphism groups?
The situation about cyclic groups of automorphisms is as follows.
If $G$ is an infinite periodic group, then its automorphism group is also infinite
(R. Baer).
If a cyclic group $A$ is the ...
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The stabilizer of a cyclic group over a group algebra
The answer is $c_{k,l}=$$\delta_{k,l}$, i.e. $a'_l=a_l.$ This is the inversion formula for discrete Fourier transform. You can easily (re-)prove it, using that for any integer $r>1,$ the sum of the ...
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