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

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-1
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2answers
12 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 ...
2
votes
2answers
41 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 ...
1
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0answers
17 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 ...
1
vote
2answers
12 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 ...
0
votes
3answers
9 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 ...
0
votes
1answer
20 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?
2
votes
0answers
17 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 ...
1
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0answers
30 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
votes
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
votes
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 ...
1
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0answers
13 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 ...
1
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0answers
33 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 ...
0
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0answers
25 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 ...
1
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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 ...
1
vote
1answer
23 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 ...
0
votes
1answer
42 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 ...
1
vote
3answers
36 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 ...
1
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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 ...
0
votes
1answer
18 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 ...
1
vote
2answers
44 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 ...
1
vote
1answer
32 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
votes
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
votes
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, ...
2
votes
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
79 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.), ...
0
votes
2answers
40 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.
3
votes
0answers
35 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 $G$?
1
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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
vote
1answer
23 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 ...
0
votes
1answer
33 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 ...
-3
votes
1answer
39 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
0
votes
1answer
36 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 ...
0
votes
1answer
18 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 ...
1
vote
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
votes
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 ...
2
votes
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 ...
-2
votes
1answer
50 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 ?
1
vote
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 ...
0
votes
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
votes
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 ...
2
votes
1answer
36 views

Showing a Group $G$ is not Simple [duplicate]

Let $G$ be a finite group of order $pq$, where $p,q$ are distinct prime numbers. Show that $G$ is not simple. Here is my attempt: $|G|=pq$. If $G$ is not simple, then it has non-trivial subgroups, ...
1
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2answers
32 views

To finde the center of $D_4$

is there a nice/smart way to find the center of $D_4$? rather then going through every element?
1
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0answers
24 views

Equations in group representation theory

I am reading this book by Margenau and Murphy: "The Mathematics of Physics and Chemistry". I think that there is a typo: In the 2nd edition, in the chapter on Group Theory, page 568, in equation ...
0
votes
1answer
22 views

Transfer homomorphism for abelian group.

If $G$ is abelian and $H \leq G$ of index $n$ , then show that transfer map is just $g \to g^n$. If i follow the definition from issac, transfer map will be same as pretranfer map as G is abelian ...
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0answers
39 views

Can we say that $A$ is a complement for a group $G$?

Let $A$ be a frobenius complement for a $G$ i.e. $A$ act on $G$ by automorphism s.t. $C_A(g)=e$ for all nonidentity $g$. Now, Action of $A$ can be linearly extended so that $A$ act on $F[G]$. As a ...
0
votes
2answers
31 views

System of generators and surjective homomorphism

Let $G$ and $G'$ be groups and $\varphi:G\to G'$ a group homomorphism and $(g_s)_{s\in S}$ an indexed collection of elements of $G$ and is also system of generators of $G$. If $\varphi$ is surjective ...
0
votes
2answers
28 views

Describe the cosets

$G$ is a cyclic group. $H$ is a subgroup of $G$. $|G|=12$, $|H|=3$. Why the sets of left and right cosets is a $H$, $xH$, $x^2H$, $x^3H$?
3
votes
2answers
66 views

A proof for $\widehat{\Bbb Z_{p^\infty}}\cong Z_p$

According to wikipedia, the Pontryagin dual of a Prüfer group is isomorphic to a group of p-adic integers. Where can I find a proof for it on the internet?
1
vote
2answers
30 views

Showing that $|N \cap Z(G)| > 1$ for normal subgroups of p-groups

I have a finite $p$-group $G$ and a normal subgroup $N$ which is not the trivial subgroup. I am asked to show that $|N \cap Z(G)| > 1$. There has been a similar question on MSE here: How to show ...