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

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If $n|m$ prove that the natural surjective projection $\pi: \mathbb{Z_m} \rightarrow \mathbb{Z_n}$ is also surjective in units

Not sure if this is the right path: since $n|m$, then if we factor $n = p_1^{\alpha_1}p_2^{\alpha_2}\ldots p_k^{\alpha_k}$ and $m = q_1^{\beta_1}q_2^{\beta_2}\ldots q_r^{\beta_r}$, then we have the ...
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2answers
15 views

Intersection of all p-Sylow subgroups is normal

Let $G$ be a finite group, $p$ a prime number that divides $|G|$ and $O_p(G)=\bigcap_{P \in Syl_p(G)}P$. Prove that 1) $O_p(G) \lhd G$ 2) $O_p(G)$ is maximal among the normal $p$-subgroups of $G$. ...
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0answers
6 views

Finding a dense $\{(f(x),g(x))\mid x\in \Bbb Z_{p^\infty}\}$ in $\Bbb T^2$

Let $p$ be a prime number. I'm trying find a preferably elementary proof for this proposition: There are homomorphsims $f,g:\Bbb Z_{p^\infty}\to \Bbb T$ such that $\{(f(x),g(x))\mid x\in \Bbb ...
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1answer
26 views

example for permutizer group

permutizer of a subgroup H of G is defined to be the subgroup generated by all cyclic subgroups of G that permute with H. You can help us give an example?
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2answers
21 views

If H and K are finite subgroups of G (another proof )

I have a question and it's solution , but I want another proof if there exist . Thanks
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1answer
25 views

number of groups with fixed number of conjugacy classes [duplicate]

why is there only finite number of (finite or infinite)groups with a fixed number of conjugacy classes? I know this is classical ,so plz give me a reference if you have. thank you
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29 views

Permutation group with two orbits of equal size contains a permutation with no cycles length 1 [on hold]

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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56 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 ...
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45 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
7 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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39 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
41 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? [on hold]

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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27 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
21 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 $p$ ...
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1answer
35 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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$\mathbb Z_{p^2}$ 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
49 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
24 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
12 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
23 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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22 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 $n$? [duplicate]

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 ...
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
41 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
16 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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35 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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31 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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19 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
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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
45 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
15 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
42 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 ...
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1answer
42 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 ...
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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 ...
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3answers
92 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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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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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 $\mathbb{Z},\mathbb{Z}\times\mathbb{Z_2},D_\infty$. ...
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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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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
20 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 = ...
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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 ...
2
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0answers
37 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 ...