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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Is a finite monoid with left cancellation property always a group?

I need to answer and show if a Monoid with left cancellation property always a group. I managed to show that it is correct when cancellation property holds for both left and right (that was part a of ...
3
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16 views

Help in step of the proof of Burnside's $p^aq^b$ theorem (Doerk-Hawkes book)

I'm reading the proof of the $p^aq^b$ Burnside's theorem from the book Finite soluble groups by Doerk and Hawkes. The fifth step of the proof says 2.5. Let $M$ and $H$ be maximal subgroups of $G$ ...
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31 views

For finitely generated free abelian groups $A,B$ if there is an onto homomorphism $A \to B$, then $\operatorname{rank}(A) \geq \operatorname{rank}(B)$

$\newcommand{\rank}{\operatorname{rank}}$For two finitely generated, free abelian groups $A,B$ prove that if there is an onto homomorphism $A \rightarrow B$, then $\rank(A) \geq \rank(B)$ Assume that ...
4
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11 views

Infinitely iterated square roots in groups

Let $G$ be a group. What are possible conditions on $G$ to ensure that there is no sequence $\{g_i\}_{i\in\mathbb Z}\subset G\backslash\{1\}$ such that $g_{i+1}=g_i^2$ for all $i\in\mathbb Z$? Does ...
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1answer
14 views

Cyclic Groups - $a^k = e \text{ iff } n|k$

I saw this proof in the book on Abstract Algebra. Here is part of it: Let $G$ be a cyclic group of order $n$ and $a$ is the generator of $G$. Then $a^k = e \iff n|k$ Proof: Suppose $a^k=e$. By the ...
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26 views

Prove that the center of a group with $385$ elements has an element of order 7. [duplicate]

Prove that the center of a group with $385$ elements has an element of order 7. By Cauchy's theorem I know that if I prove that $Z(G)$'s order is divisable by 7, we're done. So now I need to rule out ...
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10 views

The conjugate closure of a subset is a kernel of a permutation representation associated to a group action

Let $G$ be a group and $H$ be a subgroup of $G$. Let $A$ be the set of all left cosets of $H$ in $G$. We know that $G$ can acts on $A$ by $$g\cdot xH=gxH.$$ For any $g\in G$, define that ...
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16 views

Application of Bruhat decomposition in $\mathrm{GL}(n,\mathbb{F}_p)$

Consider the General Linear group $\mathrm{GL}(n,\mathbb{F}_p)$ over the field of order $p$, $B$ denote the subgroup consisting of upper triangular matrices, and $W$ denotes the subgroup consisting ...
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1answer
26 views

How to prove thar O(Ng) | O(g)

I have this exercize: $G$ is a group. $N\subset G$. Need to prove that: $$o\left(Ng\right)\mid o(g)$$ where $Ng\in G/N$. For now, without using the canonic homomorphism $\tau \left(g\right)=Ng$. ...
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2answers
39 views

Homomorphisms from $\mathbb{Z}_p$ to $\mathbb{Z}_3$

For which odd values of $p$ can we find a non-trivial homomorphism from $\mathbb{Z}_p$ to $\mathbb{Z}_3$ ? Is there any method to find those homeomorphisms explicitly? I have no any idea to handle ...
2
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2answers
158 views

Group Action as permutations

I'm trying to study on Group actions. the paper says(if I understand) that if I have Set $S$ and an action $\alpha$: $G \times S \rightarrow S$ . then the action may be viewed as permutation by $x ...
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0answers
27 views

Condition under which $HK$ is a subgroup

Suppose $G$ is a finite group and $H$, $K$ are subgroups. $H < N_G(K)$ is a sufficient condition for $HK$ to be a subgroup, but is is possible that $HK$ is a subgroup although neither $H$ nor $K$ ...
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1answer
35 views

Are subgroups of order $p^{n-1}$ maximal?

Let $G$ be finite p-group of order $p^n$, I know all maximal subgroups of order $p^{n-1}$ Is it right to say all subgroup of order $p^{n-1}$ are maximal subgroups? If not, what property $G$ must have ...
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1answer
20 views

Set invariant under reflections is a ball?

Say that $A\subset \mathbb{R}^n$ is measurable and of positive, finite measure. I'm trying to see if the following is true. If $A$ is invariant under all orthogonal reflections across $(n-1)$ ...
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18 views

Enumerating functions modulo action on both the domain and codomain.

Let $Hom(A,B)$ be the set of functions from a finite set $A$ to a finite set $B$ and let $G_A \leq S_A$, $G_B \leq S_B$ be a subgroups of the permutation groups of $A$ and $B$. For $f,g \in Hom(A,B)$ ...
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2answers
26 views

Third isomorphism theorem, about quotients

I'm trying to understand the third isomorphism theorem statement, specifically the one in Wikipedia: https://en.wikipedia.org/wiki/Isomorphism_theorem#Third_isomorphism_theorem I'm stuck at the point ...
3
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30 views

Passage to fixed point spaces is object function of a contravariant functor?

Let $X$ be a $G$-space. What is the easiest way to see that that passage to fixed point spaces, $G/H \mapsto X^H$, is the object function of a contravaraint functor $X^{(-)}: \mathscr{O}(G) \to ...
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24 views

Suppose $ S = HF $ for $ H \leq S $ and $ H \cap F = 1 $. If $ N \leq H $, show $ N \leq Z(S) $.

Let $ G $ is a soluble group and $ N $ minimal normal subgroup of $ G $. Let $ F = Fit(G) $. Suppose $ S = HF $ for $ H \leq S $ and $ H \cap F = 1 $. If $ N \leq H $, show $ N \leq Z(S) $. I show ...
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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 ...
1
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1answer
13 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 $, then every chief factor ...
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$\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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1answer
54 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
16 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
17 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, ...
3
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29 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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22 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 ...
2
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1answer
38 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
57 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
37 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
60 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
47 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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44 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 ...
1
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1answer
25 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
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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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$ 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
35 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
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 ...
0
votes
1answer
22 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
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
37 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 ...
3
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1answer
36 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
183 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
49 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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votes
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
33 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 $f_1(q)=Q$ ...
1
vote
1answer
27 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
votes
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 ...