# Tag Info

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• 587

### 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

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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### 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 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 ...