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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23 views

On formula $\sum_{i=1}^n 1/(G : H_i) = 1$ on a group $G$

Let $G$ be a group. Let $H$ be a subgroup of $G$ such that $(G : H) \lt \infty$. Then there exists a sequence of elements $a_1,\cdots, a_n$ such that $G = \bigcup_{i=1}^n a_iH$ is a disjoint union. ...
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33 views

Is there a link between music theory and the mechanics of the universe? [on hold]

The production(formation)[death] of a chromatic(spherical)[gravitational] piece(droplet)[star] of music(liquid)[space/time] minimizes the tonal-area(surface-area)[dimensions] which is the ...
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1answer
15 views

Can the order on an ordered, cancellative monoid be extended to its Grothendieck group?

Suppose we have an ordered, cancellative monoid and we wish to apply the Grothendieck group construction to it. Can the total order be extended to the larger group? Example: consider the ordered ...
-4
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1answer
49 views

I need this book by Michael Weinstein, Between nilpotent and solvable [on hold]

I need this book by Michael Weinstein, Between nilpotent and solvable. I live in iran and i can't find this book. Please help me.
2
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0answers
34 views

$ p $ the largest prime factor of $ \vert G \vert $. I show if $ p > 3 $ and $ P \in Syl_{p}(G) $ then $ P \unlhd G $.

Let $ G $ be a soluble group, $ p $ the largest prime factor of $ \vert G \vert $. I show if $ p > 3 $ for prime $ p $ and $ P \in Syl_{p}(G) $ then $ P \unlhd G $. For proof i employ the ...
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2answers
27 views

Need to prove that the A4 group is Normal sub-group of S4

I already proved that N, wich is a sub-group of S4 (4-permutations), which is all the permutations, which look's like: $(a,b)(c,d)$ (which are defintly are in A4 (even permutations of S4)) are a ...
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0answers
16 views

Endomorphisms of $\mathbb C^{\times}$

What are all continuous multiplicative endomorphisms of $\mathbb C$? As $\mathbb C^{\times} \simeq \mathbb R_{>0}^{\times}\times S^1$ this question can be reduced to the description of continuous ...
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1answer
17 views

Is Bs(1,-1) linear?

I would like to prove that the Baumslag-Solitar group $BS(1,-1)=\langle a,b| bab^{-1}=a^{-1}\rangle$ is embeddable in $GL_n(\mathbb{Z})$ for some nonnegative integer $n.$ So i tried to find two ...
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1answer
15 views

$ G $ is $ p $-supersolvable group . $ Q \in \operatorname{Syl}_{2}(G^{\prime}) $. Show $ Q \unlhd G $.

Let $ G $ is a finite $ p $-supersolvable group for odd prime numbers . Suppose $ Q \in \operatorname{Syl}_{2}(G^{\prime}) $. Now i'll show $ Q \unlhd G $. Since $ G $ is $ p $-supersolvable, then $ ...
4
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1answer
32 views

Showing a subset of $S_n$ is a subgroup

Let $P$ be the set of all the elements of $S_n$ which can be written as $\sigma\mu\sigma^{-1}\mu^{-1}$ for $\sigma, \mu \in S_n$. Show this is a subgroup. This doesnt seem to be as simple as ...
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1answer
24 views

If G is finite p-group then $d(G)=d(\frac{G}{\Omega_1(Z(G))})$

Let $G$ be a finite p-group such that $G$ has no non-inner automorphism of order p leaving Φ(G) elementwise fixed If $\Omega_1(Z(G))\le G'\le \Phi(G)$ how we can get $d(G)=d(\frac{G}{\Omega_1(Z(G))})$ ...
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1answer
33 views

Is the free abelian group of rank 2 linear?

Is the group $\mathbb{Z}^2$ linear? By linear I mean There is a injective homomorphism from $\mathbb{Z}^2$ to $GL_n(\mathbb{Z})$ for some nonnegative interger $n.$ I tried the following homomorphism ...
2
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0answers
26 views

product of subgroups and group G

Is there any example of two subgroups H and K of G whose product give G i.e. G = HK but none of which is normal in G
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2answers
174 views

Prove that G is a group

The exercise is: Let $g\in G$. $G$ is a group. Prove that $G=\{gx:x\in G$}. I know the the definition of group but the proof that is in the book is the next one: Let $H=\{gx:x\in G\}$ ...
4
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1answer
47 views

Prove that : $n \mid \varphi (a^{n}-1)$ $a>1$

Prove that : $n \mid \varphi (a^{n}-1)$ $a,n$ positive integers wih $a>1$ I know that $a$ has multiplicative order $n$ in the ring of integers modulo $a^{n}−1$ and the order of the group of ...
-1
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1answer
27 views

Order of Automorphis group [on hold]

Let $ G $ is a finite solvable group and $ N $ be a normal minimal subgroup of $ G $ that $ G = MN $ for maximal subgroup $ M $ of $ G $, which $ M \cap N = 1 $. Let $ \vert N \vert = 4 $. Then $ ...
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0answers
16 views

looking for example of infinite p-group of nilpotency class 2

Is there infinite p-group of nilpotency class 2? If p=2 or p=3 would be better. and I prefer simple examples
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0answers
28 views

Kernel of homomorphism

Let $H:=\mathbb{Z}*\mathbb{Z}/n\mathbb{Z}=\langle p,q| q^n=1\rangle.$ I wanna show that the following homomorphisms $f_1$ and $f_2$ defined by $f_1: H\to GL_n(\mathbb{Z})$ $f_1(p)=P$ and $f_1(q)=Q$ ...
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1answer
25 views

Inner automorphisms Inn(D4).

We need to show that elements of $Inn(D_4)$ are distinct , where , $Inn(D_4)= \phi_{{R_0}} , \phi_{{R_{90}}} , \phi_{H} , \phi_{D}$. Is it sufficient to construct a Cayley table for the elements of ...
2
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1answer
20 views

invariant properties between p-group and it's automorphism

Let $G$ be a p-group and $Aut(G)$ be group of automorphisms of $G$ which properties of $G$ can help us with studding $Aut(G)$? for example If $G$ is infinite/finite does this guaranty $Aut(G)$ be ...
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1answer
27 views

Endomorphism structure of the Klein four-group

I am reading the Algebra by Grillet, this is ex 17(-18), pag. 22. I understand that $V$ can be viewed as a two dimensional vector space over $\mathbb{F}_2$. Noticed this, it is easy to see that ...
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1answer
21 views

Continuous Group of Transformations

I read in chapter 2 "Weisner Method" in the book "Obtaining Generating Functions" by Elna Browning McBride In the Introduction,I did not understand the meaning of this statement " The method is ...
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29 views

Visualising a 1-(50,15,15) design.

The problem I have is the visualisation of a 1-(50,15,15) design. That is a set of 50 points and 50 blocks (lines), so that each point is on 15 lines, and each line contains 15 points. My attempts ...
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1answer
30 views

Direct product of quotient groups

Let $ G $ is a finite solvable group, Suppose $ H $ and $ N $ are minimal normal subgroups of $ G $. Then $ G/N \times G/H \cong G/N\cap H $ ?
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3answers
56 views

Find groups that contain elements $a$ and $b$ such that $|a|=|b|= 2$ and $|ab|=5$

Find groups that contain elements $a$ and $b$ such that $|a|=|b|= 2$ and $|ab|=5$ My thoughts: $|a|=|b|=2\implies a^2=e$ and $b^2=e$ I see that the group cannot be abelian as the order wont be ...
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1answer
21 views

$ N_{1} $ , $ N_{2} $ are minimal normal subgroups of $ G $, $ G/N_{1}\cap N_{2} \cong G $?

Let $ G $ is a finite group and $ N_{1} $ , $ N_{2} $ are minimal normal subgroups of $ G $ that $ N_{1} \neq N_{2} $. Suppose $ G/N_{1} $ and $ G/N_{2} $ are supersolvable. Then $ G $ is ...
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1answer
58 views

Finite partition of a group by left cosets of subgroups

Let $G$ be a group. Suppose there exist a finite sequence of elements $a_1, \cdots, a_n$ and a finite sequence of subgroups $H_1, \cdots, H_n$ such that $G = \bigcup_{i=1}^n a_iH_i$ is a disjoint ...
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What are super-translations?

There's been a lot of news lately about a possible solution to the black hole information paradox from a presentation given by Stephen Hawking to the KTH Royal Institute of Technology in Stockholm. ...
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2answers
40 views

How can I prove that $g\cdot H \cdot g^{-1}$ is also finite and has the same number of elements that $H$?

Suppose $G$ a group and $H$ is a finite subgroup of $G$ also $\forall g \in G$ the set $g\cdot H \cdot g^{-1}=\{ g\cdot h \cdot g^{-1} : h\in H\}$, is a subgroup of $G$. Prove that $g\cdot H ...
2
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1answer
66 views

How to prove that G is a cyclic Group? [duplicate]

Suppose $G$ is a finite Abelian group and, $\forall\ n\in \mathbb{N} $, there exist at most $n$ elements in $G$ which satisfy $x^n=1$. Prove $G$ is cyclic. Thanks for your help.
3
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1answer
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When can you “extend” subgroups?

Suppose $H\le G\le G'$ are subgroups, with $H$ finite index in $G$. Can you always find a subgroup $H'\le G'$ of finite index such that $H'\cap G = H$? Of course it's trivial if $G$ is finite index ...
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0answers
21 views

$ G $ is soluble, then every properties of $ G $ is inherited by $ G/N $ ? [on hold]

Let $ G $ be a finite group and $ G $ is soluble. Suppose $ N $ be a normal minimal subgroup of $ G $. Then every properties of $ G $ is inherited by $ G/N $ ?
3
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0answers
87 views

Is there a way to visualize a group?

Is there a way to picture a group in ones head? I want to "see" the difference between abelian and non-abelian group. And if f is a group homomorphism, is there a way to see that Ker(f)=1<=>f ...
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$|G:H|=p^n$ means $O_p(H)\leq O_p(G)$?

Let $H\leq G$ (finite group) and $|G:H|=p^n$, ($p$ is a prime number) prove that: $$O_p(H)\leq O_p(G)$$ note: $O_p(G)$ defined as the intersection of all Sylow-$p$ groups in $G$ I try to prove ...
4
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2answers
24 views

$G$ be a finite group of order $n$ , $H$ be a proper subgroup of order $m$ such that $(n/m)!<2n$ ; $G$ is not simple

Let $G$ be a finite group of order $n$ , $H$ be a proper subgroup of order $m$ such that $(n/m)!<2n$ ; then how to show that $G$ is not simple ? I have proceeded by Cayley's theorem , $\ker f$ is ...
2
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0answers
16 views

Frattini subgroups and nilpotent groups: bijection? [duplicate]

I have been proved that the Frattini subgroup of a finite group is nilpotent. Now I am wondering: is the converse true? I mean, if $G$ is a finite nilpotent group, is there always a finite group ...
1
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0answers
39 views

I need the paper “ J. Zhang, A note on finite groups satisfying the permutizer condition, Sci. Bull. 31 (1986) 363–365” [on hold]

I need this paper " J. Zhang, A note on finite groups satisfying the permutizer condition, Sci. Bull. 31 (1986) 363–365", But i not found. Who can help me to find this paper ?
3
votes
2answers
56 views

Is it possible to embed $\mathbb Z^n$ inside $ \mathbb Z^m$ as a $\mathbb Z$-module for $m < n$?

Is it possible to embed $\mathbb Z^n$ inside $ \mathbb Z^m$ as a $ \mathbb Z$-module for $m < n$ ? I think it's not possible. It might be a easy problem for some of you, but I really don't ...
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1answer
16 views

Definition of $ p $-supersoluble group.

I was searching for definition of $ p $-supersoluble group but not find definition. Please help me.
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1answer
15 views

$ K/N $ be a normal subgroup of $ (G/N)^{\prime} $. Now why $ K \cap G^{\prime} $ is a maximal subgroup of $ G^{\prime} $? [on hold]

Let $ G $ be a finite group and $ N $ is a normal subgroup of $ G $. Suppose $ K/N $ be a maximal subgroup of $ (G/N)^{\prime} $. Now why $ K \cap G^{\prime} $ is a maximal subgroup of $ G^{\prime} $? ...
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1answer
27 views

raising elements of profinite groups to $p$-adic powers

Let $\widehat{F_2}$ be the profinite free group of rank 2, and let $\widehat{\mathbb{Z}}$ be the profinite completion of $\mathbb{Z}$, and $\widehat{\mathbb{Z}}^\times$ its group of units. For ...
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3answers
64 views

Find the subgroup of $GL(2,\mathbb{C})$ generated by two matrices $A$ and $B$.

Find the subgroup of $GL(2,\mathbb{C})$ generated by the matrices $A$ and $B$, where $A=\begin{pmatrix} 1 & 0\\ 0 & i \end{pmatrix}$ and $B=\begin{pmatrix} 0 & 1\\ -1 & 0 ...
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1answer
41 views

How to find all homomorphism $\delta :V_4 \to \mathbb{C}^{*}$.

How to find all homomorphism $\delta :V_4 \to \mathbb{C}^{*}$. Where $V_4$ is Kleins 4 group and $\mathbb{C}^{*}$ is multiplicative group of nonzero complex numbers.
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36 views

What is the relation between $\mathcal{G}_{12}, \mathcal{G}_{23}, \mathcal{G}_{13}$?

please help me to find this. Suppose that the number of group homomorphism from the group $G_1$ to $G_2$ is $\mathcal{G}_{12}$, from $G_2$ to $G_3$ it is $\mathcal{G}_{23}$ and from $G_1$ to $G_3$ ...
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3answers
63 views

Smallest example of a group that is not isomorphic to a cyclic group, a direct product of cyclic groups or a semi direct product of cyclic groups.

What is the smallest example of a group that is not isomorphic to a cyclic group, a direct product of cyclic groups or a semi-direct product of cyclic groups? So finite abelian groups are ruled ...
1
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1answer
17 views

Let $ G $ satisfying the maximal permutizer condition, then $ G/N $ satisfying the maximal permutizer condition ?

Let $ H $ be a proper subgroup of finite group $ G $. Then permutizer $ H $ in $ G $ is defined by $ P_{G}(H) = \langle y \in G \vert \langle y \rangle H = H \langle y \rangle \rangle $. A group $ G $ ...
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0answers
21 views

What is a conjugate weight?

The authors here write that the longest element of the Weyl group is $$w_{\max} = - id$$ except for $E_6$, $A_r$ and $D_r$ with $r$ even. There they write that $w_{\max}$ acts on a weight $\lambda$ ...
0
votes
1answer
21 views

Significance of module in groups [on hold]

What is the significance of module in groups . I mean it is used for participation of sets but what is the its use in groups.
1
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0answers
38 views

On the number of group homomorphisms from $S_n$ to $S_m$

I was studying the number of group homomorphisms from $S_n$ to $S_m$ with $n\geq m\geq 7$ in this article. I have some difficulty in understanding. First of all, why such condition is given $n\geq ...
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2answers
37 views

Suppose that $\cdot$ is associative and has an identity element. Show that an element $g \in G$ has at most one inverse

Let $(G,\cdot)$ be a group with $e$ its neutral element. For an element $g\in G$, there exists one inverse element in $G$, denoted by $g^{−1}$, such that $g\cdot g^{−1}=g^{−1}\cdot g=e$. Can this be ...