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

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Action on a group descends to an action on its factor group

Let $A$ and $B$ be groups and $N\unlhd A$ is a normal subgroup of $A$. Suppose that $B$ acts on $A$; that is, there exists a group homomorphism (not necessarily monomorphism) ...
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7 views

Representing a product of orbits as a disjoint union of orbits

Let $A$ be a finite abelian group, and let $B$ and $C$ be subgroups. In the $A$-set $A/B\times A/C$, the stabilizer of any element is $B\cap C$, so we know there is a decomposition of $A$-sets like ...
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1answer
20 views

How do I construct a nonabelian group of order 1575?

I think that it should be a semidirect product of the direct product of any two of the three groups $\mathbb Z/7\mathbb Z$, $\mathbb Z/9\mathbb Z$ and $\mathbb Z/25\mathbb Z$ and the other one. But ...
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16 views

Error in Dixon's Book?

Dixon's Book: Exercise 2.1.6: Suppose $G$ is $k$-transitive for some $k > 2$, and $N$ is a nontrivial normal subgroup of G. Show that N is $(k-1)$-transitive. But we have in Wielandt's book: ...
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2answers
19 views

Describe classes of quotient group

I hope for your help in the next task: $G=M_{2\times 2}(\Bbb Z)$ - integer matrices of order $2$. Operation - addition. $H$ - matrices of the form: $\left( {\matrix{ a & b \cr 2c & ...
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32 views

Automorphisms of symmetric groups

For $ X $ a finite set, $\operatorname {card} X\not=2$, we have $\operatorname {Inn}\operatorname {Sym} X\cong\operatorname {Sym} X $. Indeed, we have the action of $ \operatorname {Sym} X $ on itself ...
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45 views

Cancellation of Direct Product in Grp

I'm thinking to the famous problem of cancellation property in Grp, i.e: $$G_1 \times G_2 \cong G_1 \times G_3 \Rightarrow G_2 \cong G_3. $$ Clearly there are many counterexamples like $\prod_{i \in ...
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1answer
51 views

A theorem from the theory of groups

Let $K$ be a (not necessarily normal) subgroup of the group $\,G$ : $\qquad\quad K < G$ A fixed element $\,g\in G\,$ can act, from the left, on all elements of $\,G\,$, thus generating a ...
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1answer
37 views

Every nonabelian group of order divisible by 6 contains a subgroup of order 6

I have a question I was hoping for help on: Prove or disprove every nonabelian group of order divisible by 6 contains a subgroup of order 6 I would guess that this statement is true based on a ...
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1answer
35 views

Rotman sylow excercise question. Did he ommit finite?

Let $P$ be a sylow $p$-subgroup of $G$. Let $N(P) \leq H $ .Prove $H=N(H)$.Did Rotman ommit that $G$ is finite? This is excercise 4.11 in the fourth edition.
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25 views

Symmetric group and fields

Let $K$ a field and $E=K(X_1,...,X_n)$ the fraction field of the domain of the polynomials $K[X_1,...,X_n]$. 1) Show that the symmetric group $S_n$ is a group of automorphism of $E$ 2) Show that ...
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1answer
21 views

Direct product of simple non-abelian groups

Let $G$ be a group, and let $K$ be a normal subgroup of $G$ which is a direct product of simple non-abelian groups. I wanted to prove that $K=C_K(H)[K,\, H]$ for every subgroup $H$ of $G$. Is this ...
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1answer
38 views

Isomorphism of $\mathbb{Z}/n\mathbb{Z}$

I came across this statement and I can't find an argument to prove it: Every cyclic group $\mathbb{Z}/n\mathbb{Z}$ is a quotient subgroup of $\mathbb{Z}/(p − 1)\mathbb{Z}$, where $p_i$ is a prime ...
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23 views

Quotient by the intersection of maximal subgroups in a free group.

My question is (a modification of) the following: Let $N=\cap M$, where $M$ is maximal normal of finite index in the free group $G=F_2$. Then what is the group $G/N$? Currently written, this is ...
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2answers
47 views

Suppose G is a group, p is prime , Then the number of elements of G of order p is multiple of (p-1) [on hold]

I need Help . "Suppose $G$ is a group, $p$ is prime , Then the number of elements of G of order $p$ is multiple of $(p-1) $". Give me any advise or note
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29 views

Doubt related to quotients (group or ring)

I was reading some notes about ring theory and modules and I've encountered with the following isomorphism: $\mathbb (R[X]/ \langle x^3-1\rangle)/ \langle x-1\rangle \cong \mathbb R[X]/ \langle x-1 ...
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1answer
28 views

Surjections from free groups

I am stuck on the following: How do I go about finding surjections from the free group of rank 2 $\mathbb{F}_2 = \mathbb{ Z∗Z}$, to the finite group of two elements $\mathbb{Z}_2$. Also, how would ...
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29 views

infinite version of lagrange theorem [duplicate]

Let G be a group and H and K subgroups of G. If K ⊂ H ⊂ G and K has finite index in G, then prove [G : K] = [G : H][H : K]. Obviously if we know G is finite, then we are done by Lagrange Theorem. ...
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2answers
159 views

Are quotient groups unique up to isomorphism

By this post, it seems quotient groups are unique up to isomorphism. is it correct? More clearly Let $G$ be a group and let $K,N\unlhd G$ be isomorphic normal subgroups. Are $\frac{G}{N}$ and ...
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2answers
88 views

There is no homomorphism from $\mathbb{Z_8} \times \mathbb{Z_2} \times \mathbb{Z_2}$ onto $\mathbb{Z_4} \times \mathbb{Z_4}$

If such a homomorphism $\phi$ existed, then the first isomorphism theorem says that $|\ker \phi| = 2$. Since $\mathbb{Z_8} \times \mathbb{Z_2} \times \mathbb{Z_2}$ is abelian, then every subgroup is ...
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1answer
20 views

maximum number of elements of order 5 in a group of order 80

Sylow's theorem says there is either 1 Sylow 5-subgroup or 16 Sylow 5-subgroups. In each Sylow 5-subgroup, there are 4 elements of order 5. Since an element of order 5 generates a subgroup of order 5, ...
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34 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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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
32 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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40 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
29 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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53 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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1answer
61 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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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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64 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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42 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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1answer
28 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
23 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
37 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
52 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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27 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
15 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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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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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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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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34 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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41 views
+200

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 ...
0
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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 ...