Tagged Questions
9
votes
1answer
195 views
Can the order of 2 mod p be arbitrarily small (relative to $p - 1$)?
Given a prime number $p$, let $\operatorname{ord}_p(2)$ be the multiplicative order of $2$ modulo $p$, i.e., the smallest integer $k$ such that $p$ divides $2^k - 1$. By Lagrange's theorem, ...
10
votes
5answers
185 views
Applications of Character Theory
Some of the applications of character theory are the proofs of Burnside $p^aq^b$ theorem, , Frobenius theorem and factorization of the group determinant (the problem which led Frobenius to character ...
0
votes
0answers
32 views
$\langle v,\sqrt{2}v\rangle_{\mathbb{Z}}$ not a discrete subgroup of $\mathbb{R}^{2}$ [duplicate]
I got a list of exercises to do and there is one of the first exercises which I do not manage to solve.
Its statement is the following:
Let $v\in \mathbb{R}^{n}$ be a nonzero vector. Using the fact ...
3
votes
1answer
30 views
Primitve roots and congruences?
Let $p$ be an odd prime. Show that the congruence $x^4$$\equiv -1\text{ (mod }p\text{)}\
$ has a solution if and only if $p$ is of the form $8k+1$.
Here is what I did
Suppose that $x^4$$\equiv ...
2
votes
1answer
63 views
Orbits of the $Aut (G)$ for $G = (Z/mZ) ^k$
$Aut(G)$ turns out to be group of invertible $k\times k$ matrices over $Z/mZ$. It turns out that the problem is equivalent to seeing whether a $k$ vector with entries in $Z/mZ$ whose gcd is a unit in ...
1
vote
2answers
38 views
Show that r is a primitive root?
Show that if $r$ is a primitive root modulo the positive integer $m$, then $ {\overline r }$ is also a primitive root modulo m if $ {\overline r }$ is an inverse of $r$ modulo $m$.
My TA did not go ...
5
votes
2answers
123 views
When is the group of units in $\mathbb{Z}_n$ cyclic?
Let $U_n$ denote the group of units in $\mathbb{Z}_n$ with multiplication modulo $n$. It is easy to show that this is a group. My question is how to characterize the $n$ for which it is cyclic. Since ...
2
votes
3answers
83 views
All group homomorphism from $ \mathbb{Z} _m $ to $\mathbb{Z}_n $ [duplicate]
All group homomorphism from $ \mathbb{Z} _m $ to $ \mathbb{Z}_n $
How could I find every group homomorphism?
2
votes
3answers
59 views
Root of multiplicity?
Show if a is a root of multiplicity $n\geq 2\ $, then $f(a) = 0$ and $f'(a)=0.$
I was trying to learn root of multiplicity and saw this question. My TA did not go over it yet but I was wondering how ...
3
votes
2answers
55 views
Subgroups that can be proven?
How do you show all subgroups of $A_5$ have order less or equal than 12?
I know you can use this lemma:
If G is a finite group, and H does not equal G is a subgroup of G such that oG $/|/$ i(H)! ...
0
votes
1answer
54 views
If ord (g) is d and $d_0$/d then ord ($g^{d_0}$) is d/$d_0$.
Let's give $ord(g)$ the name $d$. Let $d_0$ be some integer.
Lemma: Let $ord (g)$ be $d$ and let $d_0$/$d$, then $ord$ ($g^{d_0}$) is d/$d_0$.
I think this is saying once we know that $d_0$/$d$ ...
2
votes
3answers
114 views
Prove that ord (ab)=mn if ord a=m ord b=n and (m,n)=1?
Prove if $ord(a)=m$, $ord(b)=n$, and $(m,n)=1$, then $ord(ab)=mn$.
I was reading this and was thinking how this proof would look like. I tried to do it and am not sure if this is correct. Here is ...
2
votes
0answers
73 views
Representations of $\text{GL}_2(\mathbb{Q})$
Let's say that as a representation theorist I am naively interested in representations of $G(\mathbb{Q})$, where $G$ is an algebraic group defined over $\mathbb{Q}$. For the purposes of this question, ...
1
vote
1answer
26 views
Groups $G$ of order $8$ so that $U(\mathbb{Z}/n\mathbb{Z})\cong G$ for some $n$.
I cant solve this exercise.
Find all groups $G$ of order $8$ so that $U(\mathbb{Z}/n\mathbb{Z})\cong G$ for some $n$.
I need a little help here. thanks!!!
1
vote
1answer
62 views
prove that $U_{51}$,$U_{80}$ are not isomorphic
I need prove the next result:
$U(\mathbb{Z}/51\mathbb{Z})$,$U(\mathbb{Z}/80\mathbb{Z})$ are not isomorphic.
thanks for your help!
1
vote
1answer
47 views
Groups of units: Find an explicit isomorphism $U_{35}$, $U_{39}$
I need help in the following exercise:
Find an explicit isomorphism between $U(\mathbb{Z}/35\mathbb{Z})$ and $U(\mathbb{Z}/39\mathbb{Z})$.
Thanks!
5
votes
2answers
110 views
Upper bound for the sum of the orders in a finite group
This is a cute question that I found in ML. I thought a little bit about it and couldn't complete it to a solution.
Let $G$ be a non-cyclic finite group of order $n.$ Let $S(G)=\sum_{g \in G} ...
3
votes
4answers
92 views
Non-commutative or commutative ring or subring with $x^2 = 0$
Does there exist a non-commutative or commutative ring or subring $R$ with $x \cdot x = 0$ where $0$ is the zero element of $R$, $\cdot$ is multiplication secondary binary operation, and $x$ is not ...
4
votes
2answers
118 views
How to calculate $|\operatorname {SL}_2(\mathbb Z/N\mathbb Z)|$?
the answer should be
$$|\operatorname {SL}_2(\mathbb Z/N\mathbb Z)|=N^{3}\prod_{p|N}(1-{1 /p^2})$$
But first how to prove
$$|\operatorname {SL}_2(\mathbb Z/p^e\mathbb Z)|=p^{3e}(1-{1 /p^2})$$
2
votes
3answers
115 views
Proof of Fermat's little theorem using groups
I am trying to prove Fermat's little theorem using groups. Here is my proof:
Let $p$ be a prime. Since all numbers a such that $1\leq a\leq p-1$ are relatively prime
with $p$, they form a group under ...
29
votes
2answers
377 views
Are there/Why aren't there any simple groups with orders like this?
The orders of the simple groups (ignoring the matrix groups for which the problem is solved) all seem to be a lot like this:
...
1
vote
0answers
45 views
If $x \sim U(Z_n^*)$ then $x^2(mod \; n) \sim U(QR_n)$?
Define:
$Z_n^*=\{x \in Z_n | gcd(x,n)=1\}$
$QR_n=\{x \in Z_n | \exists r \in Z_n \; s.t. \; r^2 =x\}$
How can I show that $x \sim U(Z_n^*) \implies x^2(mod \; n) \sim U(QR_n)$?
Thank you.
3
votes
3answers
134 views
Legendre symbol- what is the proof that it is a homomorphism?
I know that one property of the Legendre symbol is that it is a homomorphism. However, I have not been able to find a proof that this is the case. If someone could give me or show me to a thorough ...
0
votes
1answer
53 views
Finding the order of elements in a group?
For $x \in X$, let $[x] = \{y \in X | f(y) = f(x)\}$ and let $U_n = \{[a] \in \mathbb{Z}_n | \gcd(a, n) = 1\}$ (the group of units of $\mathbb{Z}_n$). Determine the order of $[4] \in U_{23}$.
What ...
5
votes
5answers
198 views
Is $\mathbb Z _p^*=\{ 1, 2, 3, … , p-1 \}$ a cyclic group?
In undergraduate course, the two groups which are most frequently used may be $$\{ 0, 1, 2, ... , p-1\}$$ and $$\{ 1, 2, ... , p-1\}$$ where $p$ is a prime.
The first one is a group under addition ...
3
votes
1answer
54 views
Abelian groups related to $(\mathbb{Z}/p\mathbb{Z})^*$, with order close to $p$ — analogous to elliptic curve groups, but simpler
I'm looking for a construction of a group $H$ that is "a sister" to $(\mathbb{Z}/p\mathbb{Z})^*$, in the following rough sense:
Each element of $H$ can be represented by one or a few elements of ...
1
vote
0answers
83 views
Idealclassgroup for quadratic field
I have got a question about an ideal class group, namely the group of $\Bbb{Q}(\sqrt{-185})$.
I can say the following:
I can give a representant system of the group
I can name the class number: ...
0
votes
1answer
100 views
Problem in modular arithmetic using group theory
This problem is from Herstein's Topics in Algebra.
I have to use the property that if a finite set is closed under an associative product and that both cancellation laws hold in $G$, then $G$ is a ...
6
votes
2answers
89 views
$n|\phi(a^{n}-1)$ for any a>n?
I saw the proof which goes as follows:
$a^{n} \equiv 1 \mod(a^{n}-1) $, and n is the smallest power of a such that this is true.
We also know that by Euler's Identity $a^{\phi(a^{n}-1)}\equiv ...
8
votes
2answers
94 views
Structure of the group of arithmetic functions
This question was originally posted in Elements of finite order in the group of arithmetic functions under Dirichlet convolution.
and it goes as follows: Let G be the group consisting of all ...
4
votes
2answers
184 views
Elements of finite order in the group of arithmetic functions under Dirichlet convolution.
Let $(G, ∗)$ be the group of arithmetic functions $f : N \to C$ that satisfy $f (1)\neq 0$, with group operation given by the Dirichlet product $∗$. The identity function $I$ is the identity element ...
0
votes
1answer
84 views
Group Homomorphism and Elementary Divisors
Suppose $A = \langle x\mid x^m = e\rangle$, $B = \langle y\mid y^n = e\rangle$, and $C = \langle z\mid z^d = e\rangle$ (these are cyclic groups of order $m,n,d$, respectively). Also suppose $d$ ...
1
vote
0answers
74 views
Elementary divisors of a finite abelian group
Suppose $A$ is a finite abelian group.
(a) Extract from the function $h_A(n) = |\{x \in A : x^n = e\}|$ ($n \in \mathbb{Z}$) the elementary divisors of A using the fact that for a cyclic group $C$ of ...
2
votes
1answer
96 views
Using induction to show that $(\mathbb{Z}/p^a\mathbb{Z})^{\times}$ is cyclic.
I need to show that the group $(\mathbb{Z}/p^a\mathbb{Z})^{\times}$ is cyclic for odd prime $p$ and for $a\in \mathbb{N}^+$. I have already shown that $(\mathbb{Z}/p\mathbb{Z})^{\times}$ has a ...
4
votes
3answers
66 views
Every residue class $\pmod{2^a}$ can be written as $\pm 5^r$ for some $r$
So the question is to show that every residue class $\pmod{2^a}$ can be written as $\pm 5^r$ for some $r$.
The hint is to first show that:
For $a \ge 3$, and $H$ the multiplicative subgroup of ...
4
votes
1answer
109 views
Generator of a Quotient Group
Along the same vein of another question I've posted is another question that merges group theory with number theory. I havn't taken a formal course on group theory, and is likely why I'm stumped at ...
4
votes
2answers
149 views
For what algebraic curves do rational points form a group?
For what real algebraic curves do rational points form a group ?
How does this relate to Jacobian Varieties ?
1
vote
1answer
84 views
What about the Cauchy-Frobenius-orbit-counting formula
I know the proposition that says:
Let $\lambda$ be a homomorphism from a finite group $G$ into $\mathbb{C}^{\times}$. Suppose that $G$ acts on some finite set $\Omega$ and let $M$ be the number of ...
2
votes
3answers
173 views
Show $\vert G \vert = \vert HK \vert$ given that $H \trianglelefteq G$, $G$ finite and $K \leq G$.
My problem is a variation of one in Dummit and Foote:
Let $G$ be a group and $H \trianglelefteq G$. Prove: If $G$ is finite and $[G:H] = p$, a prime number, then for any $K \leq G$, either $K \leq H$ ...
0
votes
3answers
146 views
Explain why a group cannot have only one element of order 5.
I know that a group of order 5 does not have any nontrivial factors and that you cannot factor 5 as a product of two numbers larger than 1 but I do not know where to go from here?
0
votes
0answers
138 views
homomorphic encryption
Homomorphic encryption is a form of encryption where a specific algebraic operation performed on the plaintext is equivalent to another (possibly different) algebraic operation performed on the ...
4
votes
2answers
127 views
Multiplicative group of $(\mathbb Z/p^r)^\times$
I am trying to show that the multiplicative group of $(\mathbb Z/p^r)^\times$ is cyclic. I have established that the order of this group is $p^{r-1}(p-1)$. So, to show that it is cyclic, it suffices ...
2
votes
1answer
73 views
Congruence inequality
Given $n>2$, by calculation or otherwise deduce that $5^{2^{n-3}} \neq -1 \pmod {2^n}$
Note:The problem arose when I tried to deduce $\langle5\rangle \cap \langle2^n-1\rangle=\{1\}$ in the group ...
1
vote
1answer
246 views
Solving a polynomial modulo an integer
Say I have a polynomial $F$ of degree $n$ with coefficients in $Z_m$ and I wish to find $x$ such that $F(x)=0$ (mod $m$). For instance if $F(x)=x^{2}-a$ the solution would be the modulo $m$ squareroot ...
0
votes
1answer
55 views
Number of prime divisors of the order of $E_8(q)$.
I am trying to compute the number of prime divisors of the order of $E_8(q)$. I am interested in the general solution, but in particular, my problem calls for $q=p^{15}$ (for prime $p$) and $q\equiv ...
2
votes
2answers
82 views
Book about ergodic theory, group actions and number theory.
Does anyone Know about an introductory book showing the intersection between ergodic theory, group actions and number theory? I have been looking for but it has been impossible to me.
Thanks.
1
vote
2answers
319 views
The $p$-adic integers as a profinite group
How to prove that if $\mathbb{Z}_p$ is the set of $p$-adic integers then $\displaystyle{\mathbb{Z}_p=\varprojlim\mathbb{Z}/p^n\mathbb{Z}}$ where the limit denotes the inverse limit?
$\mathbb{Z}_p$ is ...
1
vote
3answers
108 views
Prove $\text{ord}(x,p^b) = p^{b-1} \cdot \text{ord}(x,p)$
Let $a=\text{ord}(x,m) : $ $a$ is the minimum value for which $x^a \equiv 1\pmod{m} $.
By inspection it appears that
$$\text{ord}(x,p^b) = p^{b-1} \cdot \text{ord}(x,p)$$ where $x,p,b,$ belong to ...
1
vote
1answer
44 views
Generating a number of a specific order
Here is what I have:
select $p$ such that $p - 1$ has a large prime factor $t$: $p - 1 = tu$, where $u$ is a random number
$n = p^2 q$, where $q$ is prime
pick random $g < n$ and compute $g_p = ...
3
votes
4answers
168 views
Subgroups of $\mathbb{Z}$?
I have this set:
$$I_a = \{ax+(7-a^2)y : x, y \in \mathbb{Z}\}$$
with $a$ integer. I have two tasks over it:
1) Prove that $I_a$ is subgroup of $\mathbb{Z}$;
2) Fully characterize this subgroup.
...
