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- For what $n$ is $U_n$ cyclic? - Mathematics Stack Exchange
17 Un U n is cyclic iff n n is 2 2, 4 4, pk p k, or 2pk 2 p k, where p p is an odd prime The proof follows from the Chinese Remainder Theorem for rings and the fact that Cm ×Cn C m × C n is cyclic iff (m,n)= 1 (m, n) = 1 (here Cn C n is the cyclic group of order n n)
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Q A for people studying math at any level and professionals in related fields
- modular arithmetic - Prove that that $U (n)$ is an abelian group . . .
Prove that that $U(n)$, which is the set of all numbers relatively prime to $n$ that are greater than or equal to one or less than or equal to $n-1$ is an Abelian
- Prove that the sequence (1+1 n)^n is convergent [duplicate]
I know the proof using binomial expansion and then by monotone convergence theorem But i want to collect some other proofs without using the binomial expansion *if you could provide the answer w
- Prove that the order of $U (n)$ is even when $n gt;2$.
Right! I like that: $ (n-1)^2=n^2-2n+1=n (n-2)+1 \equiv 1 (\bmod {n})$ I was skeptical of the line "However, we know (I forgot the theorem's name) that the number of elements of order 2 is divisible by ϕ (2)=1 " I could replace it with "Since $ (n-1)^2 \equiv 1 (\bmod {n})$, then there is at least one element of order $2$ in $|U (n)|$ " What do you think about the proof with that change?
- Proof that $U (n)$ is connected - Mathematics Stack Exchange
Thanks for the link @muzzlator I've just had a look at it and it's very interesting (and seems a lot simpler), however it uses methods a little different to those that I have been using for the course I am studying
- $\operatorname {Aut} (\mathbb Z_n)$ is isomorphic to $U_n$.
(If you know about ring theory ) Since $\mathbb Z_n$ is an abelian group, we can consider its endomorphism ring (where addition is component-wise and multiplication is given by composition) This endomorphism ring is simply $\mathbb Z_n$, since the endomorphism is completely determined by its action on a generator, and a generator can go to any element of $\mathbb Z_n$ Therefore, the
- Prove that if $\sum {a_n}$ converges absolutely, then $\sum {a_n^2 . . .
I'm trying to re-learn my undergrad math, and I'm using Stephen Abbot's Understanding Analysis In section 2 7, he has the following exercise: Exercise 2 7 5 (a) Show that if $\sum {a_n}$ converges
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