A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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Automorphism of the subgroup [duplicate]

Let $G$ a finite group and $H<G$. Let $\varphi:H\rightarrow H$ be a nontrivial automorphism of $H$. My question is: it is possible to construct a nontrivial automorphism of $G$? I had tried to ...
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1answer
22 views

Every subgroup of finite index contained in an infinite group $G$ contains a normal subgroup of $G$. [duplicate]

Let $H<G$ such that $H$ has finite index in the infinite group $G$. Then $G$ contains a normal subgroup of finite index contained in $H$. Can I create a subgroup of index $2$ in $G$ using elements ...
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Finding the orbits of the orthogonal group $O(n)$ on $\Bbb R^n$

Let $O(n)=\{M\in GL_n(\mathbb{R}):MM^t=M^tM=I\}$ an orthogonal group. I need please an explain why each orbits consists of all vectors with the same length. I know that an orbit is defined by ...
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1answer
20 views

Image and Kernel of composition of two homomorphisms

I have just showed that the composition of $a * b$ of two homomorphisms $a,b$ is a homomorphism. However, what can I say about the image and kernel of $a*b$, in terms of $a$ and $b$? Is there ...
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2answers
27 views

Question about normal subgroups and conjugacy

Is the following true? I would prefer if a hint can be provided rather than a full solution. Let H be a subgroup of the group G. If, for a fixed $g \in G\setminus H$ and a fixed $h_1 \in ...
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1answer
51 views

$|G| = 12p$ with $p>5$ is not simple

A group of order $12p$, $p>5$, is not simple, where $p$ is prime. [Hint: Consider three cases : $p=7$, $p=11$ and $p>11$.] My attempt : Let $G$ be a group. $|G| =12p$. Note that ...
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22 views

SU(N) tensor product decomposition

Let's consider the group SU(N). The adjoint representation is $\textbf{Adj}= $ $\textbf{N}^2\textbf{-1}$. The following decomposition holds generally ( have a look at this ref ) $$ ...
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26 views

Condition for appearance of singlet in product of two irreps.

By inspecting tables for tensor products of two finite-dimensional irreps of common Lie groups, I've noticed that a trivial subrepresentation only appears when the two irreps are conjugate of ...
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1answer
23 views

There are finitely many $n$-dimensional wallpaper groups

There are $7$ Frieze groups, $17$ wallpaper groups and $230$ space groups, which are the 1,2,3-dimensional cases of isometries on $\Bbb R^n$ (I think? Frieze groups seem to require $[0,1]\times\Bbb R$ ...
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1answer
59 views

Extending a given element of a free abelian group to a basis

Let $V$ be a free $\mathbb{Z}$-module (or a free abelian group) with basis $(v_1$, $v_2$, $v_3)$. Denote by $x$ an arbitrary nonzero element of $V$, say $x=mv_1+nv_2+kv_3 \in V \setminus \{0\}$; here ...
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Automorphisms of group von Neumann algebras

I study group von Neumann algebras $L(G)$, and I extremely want to know about automorophism (groups) of these algebras. Is there any good reference about this? I appreciate of everybody help.
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52 views

The cohomology of $\mathrm{GL}_n$ over an algebraically closed field

How does one go about computing the cohomology groups $H^*(\mathrm{GL}_{\kern{0.1em}{m}}(\overline{\mathbb{F}}_p),M)$? I am particularly interested in the case when $M$ is an algebraic representation. ...
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1answer
47 views

Is the following equation equivalent to the 2-cocycle condition?

Given a finite abelian group $G$, I'm looking for functions $\rho:G \times G \to U(1)$ such that 1) $~~~~~\rho(g,e) = 1 = \rho(e,g)$, where $e\in G$ is the unit element and such that 2) ...
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2answers
36 views

Difference between conjugacy classes and subgroups?

I am studying Group theory and Im not sure I understand the difference between a conjugacy class and subgroup?
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1answer
42 views

Trancendental extension Galois group

Let $K$ be a field and consider the extension $K(X)$ of rational functions with coefficients in $K$. It is common knowledge that $\text{Gal}(K(X)/K)$ is isomorphic to the group ${PL }_2(K)$, which is ...
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1answer
49 views

Group generated by $x , y$ is non-commutative when $x^2 \cdot y^{-3} = I$.

The problem: Suppose group $G$ with generators $x$ and $y$ is defined by the relation $x^2 \cdot y^{-3} = I$. It is necessary to show that the group is non-commutative. I failed to solve the ...
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1answer
68 views

Is it possible to say that if a A is a set of elements of a semigroup S, there exists some other semigroups on the alphabet A?

Suppose we have a finite semigroup, its element's set is A = {1,2,...n}, can we get some other semigroups whose element's set is A too ?
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31 views

Linear represenation of a group(can be infinite also)

Let G be a group and let $\sigma :G \rightarrow GL(V) $ be a representation of G. Assume $\sigma$ is reducible. That is $\sigma=\sigma_1 \oplus \sigma_2\oplus .... \oplus \sigma_k $ or interms of G ...
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1answer
57 views

Any nonabelian group of order 12 is isomorphic to A4, D6, or Z3 X Z4 [closed]

Can someone show me the proof for : Any nonabelian group of order 12 is isomorphic to $D_6$, $A_4$, or $\mathbb{Z}_3 \times \mathbb{Z}_4$ Ive seen a few proofs where this is included in also ...
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1answer
33 views

A finite cancellative monoid is a group? Need help seeing why the following is a false proof.

I proved this in the following way is our exam, but got $0$ out of $7$ points for it, however I fail to see why its wrong: Let $a\in S$ be arbitrary. Since $S$ is finite, there has to be some $m ...
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1answer
46 views

Square of order of a Sylow p-subgroup in the nonabelian simple groups

Is it true that for all Sylow subgroups $P$ of a nonabelian simple group $G$ that $|P|^2 < |G|$? If $P$ is abelian, this is an easy consequence of Brodkey's theorem (Suppose that a Sylow ...
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1answer
80 views

Is there any efficient algorithm for computing all semigroups of order n?

I found a paper, but here the author solves a bit different problem. My question is: Is there any efficient algorithm for computing all semigroups of order n?
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1answer
54 views

Show that $U_{14}\cong U_{18}$?

Because $|U_{14}|=|U_{18}|$ and both of them is cyclic and commutative, so i just need define a function $f : U_{14}\to U_{14}$ that f is homomorphism and bijective. $f(1)=1, f(3)=5, f(5)=7, f(9)=11, ...
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Let $D(G)$ the conmutator subgroup of $G$. If $H$ is a subgroup of $G$ such that $D(G)\subset H$ show $H$ is normal to $G$ [closed]

Let $D(G)$ the conmutator subgroup of $G$. If $H$ is a subgroup of $G$ such that $D(G)\subset H$ show $H$ is normal to $G$. Please, I appreciate any help, since I have some ideas, but those are ...
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54 views

Number of $2$-Sylow subgroups of $S_5$

Find the number of $2$-Sylow subgroups of $S_5$ and represent one of them. Would someone please give a hint for how to start?! I only can say that it should be an odd number dividing $5!$ (these can ...
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1answer
20 views

Direct sum and $FG$ homomorphism

Let $V$ be an $FG$-module and suppose that $$V=U_{1} \oplus...\oplus U_{r}$$ Each $U_{i}$ is an $FG$-submodule of $V$. For $v=u_{1}+...+u_{r}\in V$ and $u_{i} \in U_{i}$ Define $\pi_{i}: V \to V ...
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26 views

Difference between the set of generators and the alphabet of a free group

What do we mean by saying "a semigroup P is presented by generators and relations". Isn't it right only for the free semigroups? If it's right, we can't distinguish some two semigroups if they are ...
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68 views

Abstract Mathematics - Group theory and isomorphism

I have been trying to solve two problems, but I am stuck. Can anybody provide me with some links or theory to solve the following problems? The problems are from a study guide and the test exercises ...
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64 views

order $a$ = 5, $a^3b = ba^3$. show that that $ab = ba$.

Let $a, b$ be elements of a group $G$. Suppose that a has order $5$ and that $a^3b = ba^3$. I want to show that that $ab = ba$. Here is what I think: We know that we have $a^1, a^2, a^3, a^4, a^5 = ...
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1answer
63 views

Do these two permutations generate $A_n$?

Let $n$ be odd and not a multiple of $3$. Do the cycle $\sigma:=(1, 2, \dots, n)$ and any cycle of length $3$ generate $A_n$?
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39 views

when is a finitely generated abelian group finite?

I've been asked to show that a finitely generated abelian group G is finite iff $G/pG = \{0\}$ for some prime number $p$, and to find a group such that that is true for all prime $p$. Not really sure ...
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1answer
38 views

Are the free monoids always infinite?

It's the Wikipedia's definition of the free monoid: **In abstract algebra, the free monoid on a set is the monoid whose elements are all the finite sequences (or strings) of zero or more elements from ...
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44 views

I need help proving that a certain subgroup is a direct product of specific subgroups

Let G be a group, P be an abelian Sylow-p subgroup of G. Let $N=N_G(P)$ and assume that H is a complement of P in N which I believe means that $HP=N$ and $H\cap{P}=1$ Prove that $P = P_1 \times P_2$ ...
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23 views

If The order of $G$ is $pm$ and $p$ is prime such that If $H$ is normal to $G$ with order $p$ show that H is characteristic

I will be grateful for your help If The order of $G$ is $pm$ and $p$ is prime such that If $H$ is normal to $G$ with order $p$ show that H is characteristic
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Intuition of a theorem in Abstract groups

A theorem in Abstract groups Let $N \triangleleft G$ Then, $1\cdot $ If $H \leq G$ with $N \leq then H/N \leq G/N.$ Morever, if $N \leq K \leq G with K/N =H/N$ then ...
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Why is the Galois group of the polynomial $f(x)=x^5-x-1$ isomorphic to $S_5$?

Currently I am trying to understand why quintic (and higher degree) polynomials are not soluble by radicals, and one of the easiest examples of a quintic polynomial which has a nonsoluble Galois ...
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1answer
27 views

Normal Klein four-subgroup of symmetric group:S4

I've recently found a very interesting web portal about groups. I wanted to know about the normal subgroups of $S_4$ regarded as the rotation group of the cube. I found that one f them is the Normal ...
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1answer
34 views

Not a normal subgroup by left and right coset

If $G = \begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} a,b\in (\mathbb{R}) : a \neq 0$) and assume G is a group under matrix multpication Assume that K = ($\begin{pmatrix} s & 0 \\ 0 ...
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Geometric interpretation of length function of a coxeter group

It is about an exercise in Humphrey's Reflection groups and Coxeter groups exercise 1 section 5.6. Let (W,S) be a Coxeter system. It is assumed throughout the chapter that S is finite. Let $\sigma : ...
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Map of an element in a group to the conjugation by g

Let G be a group and suppose $g \in G$. $\varphi:G\rightarrow Aut\left ( G \right )$ $g \mapsto i_{g}$ is a Homomorphism with image $Inn\left ( G \right )$ where $Inn\left ( G \right )=\left \{ ...
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Prove that the funtion f: $G\rightarrow G$, defined by $f(x)=x^k$, $x \in G$ is a permutation of $G$

Help me with this exercise, I could not do it :( Let $G$ be a cyclic group of order $n$ and let $k$ be an integer relatively prime to $n$. Prove that the function f: $G \rightarrow G$, defined by ...
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Why is the group of unit upper triangular matrices solvable?

Let $GL_n(k)$ be the n by n general linear group over $k$, $B_n(k)$ be the subgroup of $GL_n(k)$ consisting of all upper triangular matrices, and $U_n(k)$ be the subgroup of $B_n(k)$ whose diagonal ...
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What is the group of rotations of a volleyball(pyritohedron)?

Practice test for Abstract algebra final, very stuck on this particular question.
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A group specified by Generators and Relations.

I'm confused with some terms in several definitions. Is an alphabet of a free group the same thing as a generator set of any group? If it is right, then by a given alphabet (set of generators) can be ...
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2answers
37 views

Galois groups over p-adic fields

Let $k$ be a finite Galois extension of $\mathbb{Q}_p$. Do we know what groups $G$ show up as $G=Gal(k/\mathbb{Q}_p)$, as $p$ varies?
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9 views

Quotient Groups Composition Series Solvable [duplicate]

Show that all the quotient groups in a composition series of a finite solvable group G are cyclic of prime order. I know a polynomial equation is solvable in radicals if and only if its Galois group ...
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1answer
30 views

Topology in the set of matrices

Let $M_n(\mathbb{R})$ be the set of real $n\times n$ matrices. I've proved that the map $\left \|\cdot \right \| \mapsto \left \| A \right \| :=\sqrt{\text{tr}(A^tA)}$ is a norm. Then I defined the ...
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56 views

Coset multiplication giving a well defined binary operation

Let G be a group and let H be a normal subgroup of G. Then prove that the rule of coset multiplication $(aH)(bH)$=$(ab)H$ gives a well defined binary operation on the set $G/H=(aH| a \in G)$ Can ...
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Are there $n$ nilpotent groups of order $n$ for some $n>1$?

Denote $nil(n)$ to be the number of nilpotent groups of order $n$. I checked the numbers $1<n\le 10^7$ , such that $n$ is neither divisble by $2^{11}$ nor a seventh power of a prime. None of ...
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43 views

Showing H is a normal subgroup by calculating left and right coset

If $G = \begin{pmatrix} a & b \\ 0 & 1 \end{pmatrix} a,b\in (\mathbb{R}) : a \neq 0$) and assume G is a group under matrix multpication Prove that H = ($\begin{pmatrix} 1 & t \\ 0 & ...