# Tagged Questions

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### When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$?

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$? I wrote it down in an imprecise way on purpose. The notation above is the linear algebra one: ...
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### A direct proof of $\binom{m\,p^k-1}{p^k-1}\equiv1~(\text{mod$p$})$?

In Nathan Jacobson's "Basic algebra I" the exercises 1.13.11-14 prove the following extension of (a part of) the Sylow's second theorem: If $p$ is a prime and $p^k\bigm||G|$, then the number of ...
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### Understanding the elements in groups(modulo, cyclic and other(?))

Question exactly as given on past exam: Consider the group $G=(\mathbb{Z}/15\mathbb{Z})^\times$ (under multiplication). Let $H$ be the subgroup generated by $2$ (that is, $H = \langle 2 \rangle$). ...
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### Order of some $\alpha$ in $S_4$

I want to work out the order of \alpha = \left[ \begin{align} & 1&2&&3&&4&\\&4&2&&1&&3& \end{align} \right] in $S_4$ Now when I think of ...
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### How to show a group is cyclic?

One question asking if $\mathbb{Z}^*_{21}$ is cyclic. I know that the cyclic group must have a generator which can generate all of the elements within the group. But does this kind of question ...
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### $U(\mathbb{Z}/n\mathbb{Z})$ Question

Consider the group $U(\mathbb{Z}/n\mathbb{Z})$. If $e$ is the order of the group, then $\forall a\in U(\mathbb{Z}/n\mathbb{Z})$, $a^e\equiv 1\pmod{n}$. But why is this true? Why is it that there ...
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### Is there any usage of this sum formula?

Let $\phi$ be the Euler phi function, then $$n=\sum_{d|n}\phi(d)$$ It is one of the most famous formulas in number theory. It can be generalized by using groups. Let $G$ be a group of order $n$ and ...
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### An exact sequence of unit groups

In the answer of K. Conrad to this question, he mentions a "nice 4-term short exact sequence of abelian groups (involving units groups mod a, mod b, and mod ab)" proving the product formula for ...
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### How would one find a) All the primitive characters modulo 8, b) All the non-primitive characters modulo 8?

Preferably explained in novice terms! I can start it off by having the multiplicative group modulo 8 with elements $[1], [3], [5], [7]$ and not sure where to go now. I see there is a similar question ...
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Let $\phi(n)$ be the Euler phi-function. If $a>1$ is an integer, then what is the remainder when $\phi(a^n - 1)$ is divided by $n$ in accordance with the Euclidean algorithm?
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### Calculating $x^{103} \equiv 2 \pmod{143}$

I need to find x, given that: $0\leq x \leq 143$ and $x^{103}\equiv 2 \pmod{143}$. I tried to use Euler's theorem $p(143)=120$, but it didn't help.
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### Which integers are a product of partition numbers?

What can be said about the set of positive integers representable as a product of the form $p(n_1)\cdots p(n_r)$ for the partition function $p(n)$ ? Such numbers $k$ arise as the number of distinct ...
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### Show that if $p$ is a prime of the form $p=4n+1$, then we can solve $x^2\equiv -1\mod p$(with $x$ an integer).

Show that if $p$ is a prime of the form $p=4n+1$, then we can solve $x^2\equiv -1\mod p$(with $x$ an integer). My attempt:If $p$ is a prime, then $U_p=${$[x]|1\leq x<p$} is cyclic.
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### $\psi (m)\leq \phi (m)$ or $\psi (m) \geq \phi (m)$ when $\psi (m)\neq 0$?

(This is different than If $x^m=e$ has at most $m$ solutions for any $m\in \mathbb{N}$, then $G$ is cyclic) I was trying to solve this: Let $G$ be a finite abelian group of order $n$ for which the ...
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### If p is a prime number of the form $4n+3$, show that we cannot solve $x^2\equiv -1\mod p$

Hint: Use Fermat's Theorem that $a^{p-1}\equiv 1\mod p$ if $p \nmid a$. (I have no idea, but something in group theory should help)
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### The Proof of Wilson's Theorem using the auxiliary multiplicative modulous group [duplicate]

(self answered question, thanks for the hints Derek Holt provided:-)) problem 18,section 4 chapter 2 in Herstein's abstract algebra: Using the results of Problem 15 and 16,prove that if p is an ...
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### Is $GL_2(\mathbb{Z}_p)$ a maximal compact subgroup of $GL_2(\mathbb{Q}_p)$?

Let $\mathbb{Q}_p$ be the set of all p-adic numbers and $\mathbb{Z}_p$ the set of all p-adic integers. Is $GL_2(\mathbb{Z}_p)$ a maximal compact subgroup of $GL_2(\mathbb{Q}_p)$? Thank you very ...
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### Closure of divisibility in denumerator, under sum of fractions

I have to prove that for a fixed positive integer n, the subset A of Q consisting of rationals with denumerator that divide n under addition, forms a group under addition. I just did that it's ...
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### Finite abelian unramified $p$-extension of a number field

Let $K$ be a number field. How many finite abelian unramified $p$-extensions of $K$ are there and what are their Galois groups? My feeling is, that every group $\mathbb{Z} / p^n \mathbb{Z}$ can occur ...
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### Are algebraic properties consistent among ALL types of number groups?

I'm embarking on a self study course of group theory, modular arithmetic, and other mathematics relating to cryptography. I notice that when studying modular arithmetic that they explicitly say that ...
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### Euler function and $\mathbb{Z}/n\mathbb{Z}$

I am trying to solve a very interesting problem about the ring $\mathbb{Z}/n\mathbb{Z}$ and Euler function $\phi (n)$, but i am not sure how to start, i have a few ideas, but none of them leads me to ...
### Given $G$ is a group and $a,b\in G$ and $ab=ba$. Prove…
$ab^n= b^na\;\; \forall n \in \mathbb{Z}$ I have been able to prove this for $n=0$ and for a positive integer (using induction). But for $n$a negative integer, I'm not able to prove it: $n=-m$ for ...