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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On Weisner Method

I read in chapter 2 "Weisner Method" in the book "Obtaining Generating Functions" by Elna Browning McBride In Sec 5" The extended form of the group generated by B and C " I did not understand ...
5
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1answer
29 views

Classification of all conjugacy classes of $GL_2(\mathbb{R})$, $GL_2(\mathbb{Q})$.

Give a classification of all conjugacy classes in the following groups. $GL_2(\mathbb{R})$ $GL_2(\mathbb{Q})$ My progress so far. If the characteristic polynomial splits, the matrix ...
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1answer
10 views

prove unique minimal normal subgroup of soluble group by $ P_{G}(M) > M $

Let $ G $ be a soluble group. If $ P_{G}(M) = \langle y\in G | \langle y \rangle M = M\langle y \rangle \rangle > M $for any subgroup $ M $ of prime power index in $ G $, then every chief factor ...
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7 views

$\mathbb{C}\{X\}^\chi$ a $G$-stable subspace of $\mathbb{C}\{X\}$?

Let $\mathbb{F}$ be a finite field with $q$ elements, and let $G = SL_2(\mathbb{F})$. The group $G$ acts on the set $X := \mathbb{F}^2 \setminus \{0\}$, the complement of the origin. For any group ...
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0answers
34 views

Commutative Diagram for group structure

I remember seeing once a commutative diagram that explained group structure. Where the associativity, identity element, inverse, multiplication and all was shown in a singular diagram, it is trivial ...
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1answer
14 views

Torsion-free nilpotent Group

I am looking for a short proof to the following fact: In torsion-free nilpotent group we have: an non-trivial element cannot be conjgate to it inverse. I know very little about nilpotent ...
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0answers
11 views

How to find a Sylow p-group?

I use Mathematica to do what I refer to as computational group theory. I know how, theoretically, to find a Sylow p-group. I want to know how to use Mathematica to solve this: Given a (large, finite, ...
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16 views

$\mathscr{O}(G/H, G/K) \cong (G/K)^H?$

What I am about to ask is related to the question presented here. Let $X$ be a $G$-space, where $G$ is a (discrete) group. For a subgroup $H$ of $G$, define$$X^H = \{x : hx = x \text{ for all }h ...
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16 views

Decomposition of certain representation of cyclic group by irreducible.

Let $C_{n}$ denotes cyclic group of order $n$. Let the set of real irreducible representations of $C_{n}$ can be listed as $$ \begin{cases} \{1,\xi,\xi^2 , \cdots \xi^{(n-1)/2}\}, \text{when $n$ is ...
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0answers
27 views

Why does there exist a deck transformation mapping here?

See Kevin Dong's answer here. Let $\widetilde{\gamma}$ be a path in $X_N$ based at $x_0^N \in X_N$ with $p(\widetilde{\gamma}) = \gamma$ a nontrivial path in $X$. Since the cover is normal, there ...
4
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1answer
51 views

Is there a “unique factorization theorem” for finite groups?

Sometimes it is difficult for me to understand what a group seems like. For example, the dihedral group $D_5$ is easy to visualise when I think it of as a "product" of two cyclic groups $C_2$ and ...
3
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1answer
32 views

$ S_{4} $ has a sylow tower?

A group having a Sylow tower is a finite group that possesses a Sylow tower: a normal series such that the successive quotient groups of the normal series all have orders that are powers of primes, ...
4
votes
1answer
50 views

If $|H|=112$ then $A_7\cap H \lhd H$?

I posted this because Alex Clark asked in chat and I'm not sure how to proceed. Let $G$ be a group such that it has a fixed subgroup isomorphic to $A_7$, which we denote simply by $A_7$. Let $H$ be a ...
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1answer
37 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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41 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
23 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 ...
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1answer
54 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.
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38 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
33 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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18 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
21 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 $ ...
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1answer
35 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
35 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
36 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 ...
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1answer
34 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
182 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
48 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 ...
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1answer
28 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 $ ...
2
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1answer
31 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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30 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
26 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
21 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
28 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
22 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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0answers
30 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
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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 ...
0
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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 ...
5
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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 ...
4
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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
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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
67 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
24 views

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
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$ 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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88 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 ...
7
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0answers
36 views

$|G:H|=p^n$ 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
votes
2answers
26 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
votes
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 ?