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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subgroup of $GL_{n+1}\mathbb{R}$ which is isomorphic

Describe a subgroup of $GL_{n+1}\mathbb{R}$ which is isomorphic to the group $\mathbb{R}^n$ under the operation of vector addition. I have no idea what this would look like. I would really ...
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59 views

Quaternion identity proof

If $q \in \mathbb{H}$ satisfies $qi = iq$, prove that $q \in \mathbb{C}$ This seems kinda of intuitive since quaternions extend the complex numbers. I am thinking that $q=i$ because i know that $ij = ...
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45 views

Matrix Groups dealing with $GL_n(\mathbb{R})$

Describe all elements $A \in GL_n(\mathbb{R})$ with the property that $AB = BA$ for all $B \in GL_n(\mathbb{R})$. I would really appreciate any help. I am not sure where to start. I almost want to ...
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How general are $\mathrm{Aut}$ groups and $\mathrm{End}$ rings?

For any locally small category $X$ and object $A\in X$ the set $\mathrm{Aut}_X(A)$ is a group w.r.t. composition $\circ$. For any locally small abelian category $X$ and object $A\in X$ the set ...
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Tell if parity check matrix is linearly independent

I know these are parity check matrixes of linear codes $C_1$ and $C_2$. $H_1= \begin{matrix} 1 & 0 & 1 & 0 & 1\\ 0 & 1 & 0 & 1 & 1\\ 1 & 1 & 0 & 0 ...
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Quotient Groups: Computational Implementation

For a permutation group $G$ and subgroup $H$, what is the best way to calculate $G/H = J$ computationally? If we store each permutation as a row in matrix ${\bf G}$ and similarly for ${\bf H}$, I ...
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120 views

Functors and Groups

Let $\alpha$ be a functor from the category of groups in the category of groups which assigns to every group $G$ a characteristic subgroup $\alpha (G)$ of $G$ and to every homomorphism $\theta : H ...
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61 views

A question about Hyperabelian Groups

A group $G$ is a hyperabelian group if has a ascending normal series with abelian factors. Prove that $F(G)$ is a hyperabelian group for all group $G$, where $F(G)$ is the Fitting subgroup of the ...
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63 views

find a special free subsemigroup

It is well-known theorem that for an elementary amenable group $G$, $G$ has exponential growth rate iff $G$ contains non cyclic free semigroup. Now I am interested in the following questions: Let ...
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80 views

Isomorphism of Group with the Image of the Group - Fraleigh p. 82 Lemma 8.15

I found multifarious duplicates that I listed at http://math.stackexchange.com/a/631364/53934. I edged the purple part because my answer proves it more efficiently. I remember that any function ...
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Show that a group of order $28$ contains $2$ subgroups $H_1 > H_2$ such that $|H_1|= 14$ and $|H_2| = 7$

I know that there is only one $7$-Sylow group in $G$ ($n_7= 1$). I just need to find a normal subgroup of order $2$ , I know that there is an element of order $2$ in $G$ but cannot continue.
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Group and preorder

Let $G$ be a group. Let $a, b$ be elements of $G$. We denote $\operatorname{Hom}(a, b) = \{ab^{-1}\}$. Then we get a category whose set of objects is $G$. We can regard this category as a preorder in ...
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146 views

Does there exist any non trivial finite subgroup?

Let $G$ be a group of infinite order . Does there exist an element $x$ belonging to $G$ such that $x$ is not equal to $e$ and the order of $x$ is finite?
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153 views

Naturally Ordered Semigroup: Why does axioms imply order of group is infinite countable ? Why are every group equal up to isomorphism?

Naturally Ordered Semigroup: http://www.proofwiki.org/wiki/Definition:Naturally_Ordered_Semigroup I know $\mathbb N$ is a naturally ordered semigroup by definition. Also I know that the naturally ...
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324 views

How a group of groups is called?

How is a group of groups is called ? I'd like to give you an example. Assume we have $4$ symbols: $a, b, c, d$. One group $E_1$ is $\{a\rightarrow b, b\rightarrow a\}$. Another group $E_2$ is ...
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1answer
66 views

Question about primitive group actions

In Glass' Partially Ordered Groups Corollary 7.4.4 says: If $G$ is an ordered group and $(G,G)$ is the right regular representation, then $(G,G)$ is primitive if and only if $G$ is ...
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61 views

Prove the following elementary properties of the group character

I'm reading the book appendix on group theory of the book Quantum Computation and Quantum Information by Nielsen & Chuang. I'm having trouble with exercises A2.11 and A2.12. This isn't homework ...
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53 views

About Sylow p-subgroup

Let $G$ be a group and $P$ a Sylow $p$-subgroup of $G$. I know that $P$ consists of all $p$-elements in $N_{G}(P)$. My question is the following: If $P$ is any $p$-subgroup of $G$, and if there is no ...
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77 views

Using semiproduct to construct a non-abelian group.

I want to construct a non-abelian group of order $pq$ which is a semidirect product of two cyclic groups of orders $p$ and $q$ with $p|q-1$. For example, consider the cyclic groups $C_3=\langle ...
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1answer
106 views

A problem from I.N. Herstein

Suppose a finite set $G$ is closed under an associative product and that both cancellation laws hold in $G$. Prove that $G$ must be a group. The cancellation laws are: $a\cdot$$u=a\cdot$$w$ implies ...
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How do you prove a subgroup is a normal subgroup of a group?

I have a group $G=\{f_{a,b}\mid \text{$a,b\in\mathbb R$ and $a\ne 0$}\}$ under composition of maps, where $f_{a,b}=ax+b$. I've proven that it is a group and then I have proved that ...
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83 views

A question on finitely generated abelian groups

Let $M$ be the torsion subgroup of a finitely generated abelian group $A$ with $M = 0$. Then it is not difficult to prove that there exists a $r \in \mathbb{Z}_{\geq 0}$ such that $A \cong Z^r$ . ...
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Centralizer/Normalizer of abelian subgroup of a finite simple group

My concern is to look for a classification of : Centralizers/Normalizer of elementary abelian subgroups of a simple group. Centralizers/Normalizer of abelian subgroups(Not necessarily elementary) ...
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Exercise 5C10 in Isaacs' Finite Group Theory

Problem: Suppose that $G$ is simple group and has an abelian Sylow $2-$subgroup of order $8$. Show that the order of $G$ is divisible by $7$. Is there any hint to solve this problem? I'll be glad if ...
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Subgroup of multiplicative group of nonzero real numbers $\Bbb R^*$ with index $2$

Let $\mathbb{R}^*$ denote the multiplicative group of nonzero real numbers. Is there a subgroup of $\mathbb{R}^*$ with index $2$?
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31 views

Quasinormal subgroup which is not nomal [duplicate]

I am searching for an easy example of a subgroup which is quasinormal but not normal. Please help.
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114 views

Question on wreath product

Let $H$ be a cyclic group of order n. Let $K$ be a subgroup of $S_n$. Let $K$ acts on $H^n$ as $(h_1, h_2,...h_n)^a=(h_{a^{-1}(1)},.....h_{a^{-1}(n)})$ where $a\in K$. My question is: What is $H^n ...
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785 views

Inverse function of isomorphism is also isomorphism

Let $G$ be a group, and let $p:G\rightarrow G$ be an isomorphism. Why is $p^{-1}$ also an isomorphism? We know that $p(a)p(b)=p(ab)$ for any elements $a,b\in G$. We also know $p(a^{-1})=p(a)^{-1}$ ...
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$\mathbb{Q}$ is not isomorphic to $\mathbb{Q}^+$ [duplicate]

How can we show that $\mathbb{Q}$ as an additive group is not isomorphic to $\mathbb{Q}^+$ as a multiplicative group? Both have a countable number of elements, neither is cyclic, neither has an ...
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172 views

$\mathbb{Z}$ is not isomorphic to $\mathbb{Q}$

How can we prove that $\mathbb{Z}$ is not isomorphic to $\mathbb{Q}$? Both of them have a countable number of elements, so cardinality doesn't help. $0$ is the identity and $-x$ is the inverse of $x$ ...
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Coproducts exist in the category of groups

This is a part of the proof that coproducts exist in the category of groups. I took it from the book Algebra by Serge Lang (This is how the proof starts): "Proof: Let {$G(i) | i∈I$} be a family of ...
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$xay=a^{-1}$ implies $yax=a^{-1}$?

If $x,a,y$ are elements in a group $G$ such that $xay=a^{-1}$, then is it always true that $yax=a^{-1}$? The two equations are equivalent to $xaya=e$ and $yaxa=e$, but I don't know whether these two ...
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115 views

Permutation that fix elements within set is subgroup?

(a) Let $A$ be a finite set, and $B\subseteq A$. Let $G$ be the subset of $S_A$ (permutations of $A$) consisting of all the permutations $f$ of $A$ such that $f(x)\in B$ for every $x\in B$. Prove ...
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113 views

Question about Sylow subgroup

I am Dealing with this question " Let $\rho: $G$ \longrightarrow$ $Sym(G)$ be the regular representation of the finite group $G$. Show that $\rho (G) \leq Alt(G)$ if and only if a Sylow 2-subgroup ...
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Group theory: Subgroup over finite fields

Would I be right in saying that the subgroups of $G=\mathbb{Z}_5$ are the cyclic groups $\langle 1 \rangle = G,\langle 2 \rangle,\langle 3 \rangle <4>$?
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Automorphisms of $D_{2n}$ [closed]

Let $D_{2n}$ be presented as $\langle r, s \mid r^n=s^2=1, srs=r^{-1}\rangle$. Is it the case that sending $r$ to any power of $r$ relatively prime to $n$ and sending $s$ to any reflection gives an ...
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How does Cauchy's theorem follow from Sylow's theorem?

Very quickly, Sylow's first theorem says a sylow p-subgroup of order $p^rm$ exists and Cauchy's theorem says if $p \vert |G|$ then there is an element of order $p$. It's often said that Cauchys ...
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Show that a certain function $\tilde{f}:S^3\to\mathbb{R}$ induces a function $f:S^3/S^1\to\mathbb{R}$ (group actions)

I am a bit stuck on a homework assignment and I'm hoping someone can push me in the right direction. Consider the 3-dimensional sphere: $$S^3=\{(z_1,z_2)\in\mathbb{C}:|z_1|^2+|z_2|^2=1\}$$ and the ...
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What group are the group of symmetries of these figures isomorphic to - Fraleigh p. 85 Theorem 8.23, 24, 26

In this section we discussed the group of symmetries of an equilateral triangle and of a square. In Exercises 23 through 26, give a group that we have discussed in the text that is isomorphic to the ...
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358 views

How to Visualize Diagonally Opposite Vertices

Consider a cube that exactly fills a certain cubical box. As in Examples 8.7 and 8.10, the ways in which the cube can be placed into the box corresponds to a certain group of permutations of the ...
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Holomorph is isomorphic to normalizer of subgroup of symmetric group?

Let $G$ be the holomorph of $H$, and thus the semidirect product of $H$ and $K=Aut(H)$. If $H$ has order $n$, then I have already shown that, letting $G$ act on the $n$ left cosets of $K$ by left ...
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92 views

Finite groups with unique minimal subgroup

Let $G$ be a finite group. Let $G$ has a unique non trivial minimal subgroup. Then $G$ is a p-group. How to prove the theorem which says that: If $G$ has a unique non trivial minimal subgroup and if ...
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Could a group embed normally in another group in which any automorphism is attained by conjugation?

Problem Suppose $G$ is a group. Can we always find a group $\tilde G$, such that $G$ is a normal subgroup of $\tilde G$, i.e. $G\triangleleft\tilde G$, and for each automorphism ...
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No group of following property. Is this true?

Let $p$ be a prime greater than 3 and $G$ be group of order $p^5$. Is it true that there is no group $G$ of order $p^5$ such that the order of frattini subgroup is $p^3$ and the order of center is ...
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Conjugacy classes of unipotent $\mathbb{Z}\times\mathbb{Z}$ in $GL_3(\mathbb{Q})$

Let $G=\mathrm{GL}_3(\mathbb{Q})$. Now, consider all subgroups in $G$ of the form $\mathbb{Z}\times\mathbb{Z}$ consisting only of unipotent elements (elements whose eigenvalues are all $1$). How ...
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Direct Product of more than 2 Groups

A group $G$ can be shown to be the direct product of 2 of its normal subgroups if they generate the entire group and intersect trivially. Does this extend to more than 2 subgroups? For example, if $G$ ...
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Maximal subgroups of direct product of solvable groups

Let $G_1$ and $G_2$ be finite solvable groups and $M$ be a maximal subgroup of $G=G_1\times G_2$ show that one of the following holds: $$ G_1\times\{e\}\leq M,\ \{e\}\times G_2\leq M,\ M \unlhd G.$$ ...
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Simple question regarding number of elements in cyclic subgroups

Let $G$ be a cyclic group with $N$ elements. Then it follows that $$N=\sum_{d|N} \sum_{g\in G,\text{ord}(g)=d} 1.$$ I simply can not understand this equality. I know that for every divisor $d|N$ ...
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— Cartan matrix for an exotic type of Lie algebra --

(1) Is there a notion of Cartan matrix for non-semisimple Lie algebra? For example, consider this Lie algebra: $$ [X_i, X_j] = f_{ij}{}^k X_k \qquad\qquad [X_i,Y^j] = - f_{ik}{}^j Y^k \qquad\qquad ...