The study of symmetry: groups, subgroups, homomorphisms, group actions.

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17 views

Permutation group with two orbits of equal size contains a permutation with no cycles length 1

For G a permutation group on a set X with exactly two orbits of the same size, how can we prove that there exists a permutation g in G that does not contain any cycles of length one?
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0answers
26 views

$G/Z(G) \cong \mathbb Z_p \times \mathbb Z_p$ then $p||Z(G)|$

Problem Let $G$ be a finite group with $G/Z(G) \cong \mathbb Z_{p} \times \mathbb Z_{p}$. Then $p| |Z(G)|$. My attempt at a solution Consider the action of $G$ on itself by conjugation. By the ...
3
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2answers
33 views

For a given group $G$ , what are the sets on which a non-trivial group action of $G$ can be defined ?

Say we are given a group $G$ , we want to find those sets on which we can define an action of $G$ ; now in this sense any set $X$ works as we can always define the trivial action $o:G \times X \to X$ ...
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0answers
4 views

Preimage of a natural morphism

Let $\mathbb R^n$ and $L$ be a additive subgroup of $\mathbb R^n$. COnsider the natural map: $p:\mathbb R^n\to\mathbb R^n/L$ If $X\subset \mathbb R^n$ then the preimage of $p(X)$ is $X$? Thank ...
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21 views

Permutation group of a set

If we let $G$ be a finite permutation group of a finite set $X$ and assume that $G$ has exactly 2 orbits of the same cardinality, how can we show that there is some permutation in G that has no cycles ...
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2answers
28 views

Which are the nine Sylow $2$-subgroups of $S_3 \times S_3$? What is the only Sylow $3$-subgroup of $S_3 \times S_3$? And the most important…Why?

Which are the nine Sylow 2-subgroups of $S_3 \times S_3$? What is the only Sylow 3-subgroup of $S_3 \times S_3$? And the most important... why? I am doing an independent study of Abstract Algebra. ...
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1answer
22 views

Prove the automorphism given by $\phi \left(g\right)=\left(g^{-1}\right)^t$ is not an inner automorphism of $SL_n\left(R\right)$

Prove the automorphism given by $\phi \left(g\right)=\left(g^{-1}\right)^t$ is not an inner automorphism of $SL_n\left(R\right)$ Having no success with this question, I turn for your help =] I ...
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1answer
18 views

Composition factors of linear groups

The following problem comes from an algebra exercise and since two days or so, I am not able to find a satisfying solution: Let $p$ be a prime with $p \geq 5$. Let $F_p$ denote the field with $p$ ...
2
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1answer
31 views

Proving a subgroup is normal

Problem Let $G$ be a group with $|G|=pm$, $p$ prime and $p \geq m$. Suppose there is $H$ subgroup of $G$ with $[G:H]=p$. Show that $H$ is normal. This problem was given to me in class just after ...
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2answers
15 views

$\mathbb Z_{p^2}$ is not a non trivial semidirect product.

I am trying to prove that the group $\mathbb Z_{p^2}$ (p prime) is not a non trivial semidirect product. Since a group $G \cong K \rtimes H$ if and only if for all short exact sequences $$0 ...
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2answers
46 views

For any group $G$, $|G/Z(G)| \neq 91$.

In Malik's Fundamentals of abstract algebra, one can find the following problem: Prove that for any group $G$, $\vert G/Z(G)\vert \neq 91$. This exercise is just ahead of Sylow's theorems. I've ...
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0answers
23 views

If all sylow subgroups are cyclic, prove that G is solvable

I came across a statement which I am unable to prove by myself that if $G$ is a finite group then if all its sylow subgroups are cyclic, prove that G is solvable. If it has been asked before please ...
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2answers
14 views

Order of elements in a commutative/abelian group

Prove that if $(G, ◦)$ is a (not necessarily finite) commutative group, and if $g$ and $g'$ are members of $G$ which have finite orders (say $ω$ and $ω'$ respectively), then $g ◦ g'$is of finite ...
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3answers
11 views

find the Bijective function that answers the criteria: [0,1] -> [0,1) union [3,4]

find the Bijective function that takes elements of [0,1] (the numbers between 0 and 1 included) and matches exactly one element in the set [0,1) $\bigcup$ [3,4] (notice that 1 is not defined. the big ...
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1answer
22 views

Solvability of ${\rm GL}_2(\mathbb{C})/\mu_n$.

Let $n\geq 1$ be an integer and $\mu_n$ is the group of $n$th roots of unity. Is it true that the group ${\rm GL}_2(\mathbb{C})/\mu_nI_2$ is solvable?
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18 views

To calculate what is $\sum_{1 \le m<n ;(m,n)=1} m^2$ or what is the remainder when $\sum_{1 \le m<n ;(m,n)=1} m^2$ is divided by $n$?

For an integer $n >1$ , what is the sum of the squares of all the positive integers that are less than $n$ and relatively prime to $n$ that is I am trying to calculate $f(n):=\sum_{1 \le m<n ...
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1answer
51 views

Prove that there is no isomorphism between any two of the groups $ Aut(\hat{C}) $,$ Aut(H^+) $(upper half plane) and $ Aut(C) $

Referring the groups of automorphisms (holomorphic bijections) of the respective domains. An equivalent statement would be: there is no isomorphism between any two of PSL(2,C), PSL(2,R) and ...
2
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2answers
38 views

Does (Z, +) have two generators but infinitely many generating sets?

We say the group of integers under addition Z has only two generators, namely 1 and -1. However, Z can also be generated by any set of 'relatively prime' integers. (Integers having gcd 1). I have ...
3
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1answer
38 views

Commutators in a group

Let $G$ be a group and for $x,y\in G$, define $[x,y]=x^{-1}y^{-1}xy$ to be the commutator of $x$ and $y$. If $y_1,\cdots,y_n\in G$, is it true that $[x,y_1\cdots y_n]$ can be written of a product of ...
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0answers
14 views

Nilpotent Orbits Of L(E8(2))

Is there any computational method in order to find a representative for a nilpotent orbits of the Lie Algebra L(G) , where G is a exceptional groups of Lie type E8(2).Also, How many nilpotent orbits ...
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0answers
34 views

Automorphisms of $Z_{p^{i_1}}*Z_{p^{i_2}}*…*Z_{p^{i_n}}$

If $Z_{p^{i_1}}\times Z_{p^{i_2}}\times\cdots\times Z_{p^{i_n}}=\langle a_1,...,a_n\rangle$, then each automorphism of this group is the forms as follows, $$\sigma:a_j\rightarrow ...
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28 views

Stuck in Preissmann's theorem

I am stuck on following the proof of Preissmann's theorem, whose statement is that Let $(M,g)$ be a closed connected Riemannian manifold of negative sectional curvature. Then every nontrivial ...
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0answers
18 views

How many permutations do we need before we're in $SU\left( n\right)$?

Let $\mathcal{L}\subseteq \mathfrak{su}\left( n\right)$ be a Lie algebra for $n \geq 2$ with Lie group $G = e^{\mathcal L}$, and let $X \in G$ be represented by an $n\times n$ matrix (I prefer fixing ...
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1answer
24 views

Derived subgroup of a finite non-Abelian p-group is proper?

How do I show that the derived subgroup of a finite p-group is always proper? In Abelian groups, it's trivial. In non-Abelian groups, my intuition is that there should be some way to relate G/Z(G) to ...
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1answer
43 views

Steps to construct the Field of fractions of Gaussian Integers $\mathbb{Z}[i]$ [duplicate]

i don't know how to construct such field $\mathbb{Q[i]} $ from $\mathbb{Z[i]}$. I know the following: $(a+bi,c+di)\sim (m+ni,r+si)$ iff $(a+bi)(r+si)=(c+di)(m+ni)$ is the equivalence relation and if ...
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3answers
38 views

Problem regarding proving a permutation group

The question states: Show that the set of permutations of three objects form a group. Give the multiplication table for this group. If we take three distinct objects, the set of the ...
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0answers
13 views

Conjugacy classes for automorphism group of simple lie type group

I have two questions. Thanks for any comments. Suppose $S$ is a simpe group of Lie type in characteristic $p$. Also suppose that $G=Aut(S)$. 1) Is there any reference for conjugacy class of element ...
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1answer
19 views

Groups, neutral elements and uniqueness

suppose for a group G, there is an element E that maps each element to itself, but for each element there is an infinite subset of G that maps it to itself, but if you take the intersection of those ...
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2answers
45 views

Guidance and sanity check needed - question on the isomorphism theorems

The question is from Joseph .J Rotman's book - Introduction to the Theory of Groups and it goes like this: $A,B,C$ are subgroups of $G$, so $A\leq B$, prove that if $(AC=BC\ \text{and}\ A\cap ...
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1answer
34 views

If $\lvert g \rvert=m$ is finite then prove that $ng=0$ if and only if $m\mid n$.

Let $G$ be an abelian group and let $g \in G$. If $\lvert g \rvert=m$ is finite then prove that, for $n\in \mathbb Z$, $ng=0$ if and only if $m\mid n$. I think this amounts to proving that: $$ng=0 ...
2
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1answer
40 views

What is the name of this terminology?

Let $G$ be the group generated by a set $X=\{x_1,\cdots,x_n\}$. Then each element can be (not necessarily uniquely) written as a product of the form $x_{j_1}^{e_1}\cdots x_{j_k}^{e_k}$, where each ...
0
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1answer
29 views

Normal Subgroups in Group Theory

I am quite confused about the Group Theory. In particular, would like to ask whether is this statement true. If G is a group, and H is a normal subgroup of G, then |H| * |G/H| = |G| Thanks! Also, ...
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3answers
37 views

Help to find $\frac{\mathbb{Z}\times\mathbb{Z}}{\left<(1,2),(2,3)\right>}$

I can prove that $\frac{\mathbb{Z}\times\mathbb{Z}}{\left<(1,2)\right>}$ is isomorphic to $ \mathbb{Z}$. Please help me to find ...
3
votes
3answers
85 views

Proof that $A_n$ the only subgroup of $ S_n$ index $2$.

I have what seems to me a very simple proof that $A_{n}$ is the only subgroup of $S_{n}$ of index 2. Since I've seen other people prove it with what feel like really complicated methods (Like here.), ...
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2answers
43 views

Group with topology which is not topological group

What will example of a group G with topology such that f: G to G such that f(x) = -x and g: G * G to G such that g((x,y)) = x * y (where * is binary operation on G) both are not continuous.
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0answers
36 views

Automorphism group of a topological space

Let $G$ be any group. Is there a topological space $(X,\tau)$ such that the automorphism group $\textrm{Aut}(X,\tau)$ is isomorphic to $G$?
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0answers
31 views

Properties of an infinite group with an infinite cyclic normal subgroup

Let $G$ be an infinite group with an infinite cyclic normal subgroup $H$ such that $|G/H|=2$ and is cyclic. Show that $G$ is isomorphic to one of $\mathbb{Z},\mathbb{Z}\times\mathbb{Z_2},D_\infty$. ...
1
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1answer
24 views

Proving that this mapping is one to one

Let $Q$ be the field of quotients of the Gaussian integers (integer complex numbers) and let $R$ be the the set of all complex numbers of the form $a +bi$ such that both $a,b$ are rationals I have to ...
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1answer
34 views

Isomorphism with Euler phi function

Let $m_i > 1$, where $1 ≤ i ≤ n$, be integers, pairwise relatively prime. Let $m = m_1 \cdots m_n$. Let $\phi(m)$ denote the order of the group $(Z/mZ)^×$. The function $\phi : Z_+ → Z_+$ is called ...
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1answer
41 views

Looking for non-commutative group with non-trivial center such that the quotient $G/Z(G)$ is non-commutative [on hold]

Give example of a non-commutative group $G$ with non-trivial center $Z(G)$ such that $G/Z(G)$ is not commutative , Please Help
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1answer
37 views

Quotient field of gaussian Integers

Let $D$ be the set of all gaussian integers in the from of $m+ni$ where $m,n \in Z$ Carry out the construction of the quotient field $Q$ for this integral domain.Show that this quotient field is ...
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1answer
19 views

Lattice with conditions

In Book's Algebraic number theory, I. Stewart page 142: Theorem 7.2: If $p$ is prime of the form 4k+1 then $p$ is sum of two squares. Proof: The multiplicative group $G$ of the field $\mathbb ...
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1answer
48 views

If $H \leq G, \exists g \in G$ such that $HgHg^{-1} = G$, then $H = G$

Just wanted some overall feedback from a homework question. Let $G$ be a group where $H \leq G$. Prove that if $\exists g \in G$ such that $HgHg^{-1} = G$, then $H = G$. $\it{Proof.}$ Note that $H = ...
2
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1answer
37 views

Extensions of $\mathbb{Z}/(m)$ by $\mathbb{Z}$

I know that $\text{Ext}_{\mathbb{Z}}^1(\mathbb{Z}/(n), \mathbb{Z}) \cong \mathbb{Z}/(n)$. I am trying to use this to show that the extensions of $\mathbb{Z}/(n)$ by $\mathbb{Z}$ are $$0 \to ...
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0answers
36 views

A Question on the Quotient Group and/or set of cosets

I'm just confused about a somewhat simple fact about quotient groups. If we have: $$H<G/N$$ is a subgroup of the quotient of a finite group $G$ by $N\trianglelefteq G$, and $|H|=n$. Can we ...
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1answer
51 views

The identity element of a group

We define the process in Z. Then, is a group. In this group,which is the identity element? The correct answer is the element 10. why ?
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1answer
41 views

Verification of using Fermat's Little Theorem

So I am asked to find what is $5^{102}$ in $Z_{11}$: The answer I have is as follows: $5^{10}=1$ by Fermat's Little Theorem So $(5^{10})^{10}$=$5^{100}=1$ So $5^{102}$=$5^2$=4 in $Z_{11}$ Is this ...
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0answers
15 views

Good, intuitive materials for learning Field of Quotients, Polynomial Rings and Factorization

Its approaching end of term and my instructor for Algebra just sped through these three topics, pretending we understand. Can anyone who has taken this course before recommend some books and resources ...
2
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1answer
41 views

$|G|=p^2$ then $G \cong \mathbb Z_{p^2}$ or $G \cong \mathbb Z_{p} \times \mathbb Z_{p}$

Problem Let $G$ be a group with $|G|=p^2$ for some prime $p$, then $G \cong \mathbb Z_{p^2}$ or $G \cong \mathbb Z_{p} \times \mathbb Z_{p}$. I think I came up with a solution to this problem but I ...
1
vote
3answers
58 views

Are there in pure mathematics ensembles of number's which not divided by them self except $0$?

In pure mathematics we know well that each number divided by him self except $0$ , the question that let me confused is: Is there a proof in pure mathematics show to us that there are others ...