The study of symmetry: groups, subgroups, homomorphisms, group actions.

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30 views

Show that $\mathbb{F}_p^\times \simeq \text{Aut}(\mathbb{F}_p^+) $ holds

I want to show, that $\mathbb{F}_p^\times \simeq \text{Aut}(\mathbb{F}_p^+)$ holds with $p$ prime. ($\mathbb{F}_p^+$ is the additive group, $\mathbb{F}_p^\times$ multiplicative group) As hint we ...
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1answer
33 views

Proof of the theorem that relates the size of the conjugacy class to the order of the centralizer subgroup

Could you please explain the step in this theorem and proof that I do not follow: If $G$ is a finite group, let $C_{G}(x)$ be the centralizer of $x$ in $G$, that is $C_{G}(x)=\{g \in G: xg=gx\}$ ...
1
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0answers
52 views

Prove that $\operatorname{Aut}(S_n)$ isomorphic to $S_n$ when $n\geq 3$ and $n \neq6$.

Prove that $\operatorname{Aut}(S_n)$ isomorphic to $S_n$ when $n\geq 3$, and $n \neq 6$. I can see that the automorphisms of $S_n$ have the same structure as $S_n$. But I am having trouble ...
4
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0answers
41 views

Prove this group $G$ is abelian [duplicate]

Let $G$ be a finite group and $\alpha$ be an automorphism of $G$ which fixes only the unit of $G$ (if $\alpha(a)=a$, then $a=1$). And $\alpha^2=1$. Show that $G$ is abelian. I think it is enough ...
1
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1answer
51 views

How prove this $Z(H)\neq 1$, if for any $g\in G\setminus H, H\cap H^g=1$

Let $2||H|$, and let $H$ be a subgroup of $G$, $H\le G$, such that for any $g\in G\setminus H$ the following holds. $$H\cap H^g=1$$ Show that :$$Z(H)\neq 1$$ where $Z(H)$ is center of the $H$. ...
2
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3answers
64 views

Motivation and intuition of double cosets

In Dummit and Foote, there was a problem on double cosets. Let $H$ and $K$ be subgroups of $G$ and $x\in G$, $HxK$ is the double coset of $x$. I can prove the double cosets partitions $G$. However, I ...
3
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2answers
71 views

Characterize the natural numbers $n$ such that there is a surjective group homomorphism from $S_n$ to $S_{n-1}$.

This has always been one of my most favorite exercises from Group Theory, and I was surprised to see that this hasn't been asked before. To repeat: Characterize the natural numbers $n$ such that ...
5
votes
2answers
102 views

Understanding what the Sylow theorems say about $p$-groups

I have a simple question. If we consider a group $G$ with order $p^k$ for a prime $p$. For example $125=5^3$. What we can obtain from sylows theorem? (I already understood it for the other cases, ...
2
votes
1answer
15 views

Eigenvectors of a Lie group invariant covariant matrix

Suppose you have a $n\times n$ covariance matrix $C$ that is commuting with all group elements, $g$, of a non abelian Lie group $G$, i.e. $[C,g]=0$ for all $g \in G$. Can we derive explicitly the form ...
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2answers
24 views

Group with order $|G|=p^2q^2$ and $p\not\mid q-1, p\not\mid q+1$

Group with order $|G|=p^2q^2$ and $p\not\mid q-1, p\not\mid q+1$. $q\not=p$ both prime. I want to show that there is only one $p$-Sylow subgroup. Let $S_p(G)$ the number of $p$-Sylow subgroups. I ...
2
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1answer
82 views

on primitive group actions with abelian stabilizers

I am trying to solve the following exercise from Dixon and Mortimer: Let $G$ be a finite primitive permutation group with abelian point stabilizers. Show that $G$ has a regular normal elementary ...
2
votes
2answers
31 views

Cocompact group actions have cobounded orbits

Assume $X$ is a complete, locally compact, geodesic metric space (in particular, $X$ has closed balls are compact, by the Hopf-Rinow Theorem). Assume $G$ acts isometrically on $X$. We say $Q\subset X$ ...
2
votes
1answer
101 views

Modern mathematics for dummies

I have poor university mathematical education, but Math fascinates me, so I decided to educate myself for a bit. I know there is a dozen modern mathematical fields I know nearly nothing about like ...
2
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2answers
70 views

For a transitive permutation group $G,$ show that there is some $\sigma \in G$ such that $\sigma(a) \neq a$ for all $a \in A.$

Here is a problem that I have been working on. I was able to prove part A, but am having problems with part B. Thanks! Let $G$ be a permutation group acting on a finite set $A.$ If $g\in G,$ let ...
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2answers
64 views

Use Sylow's theorem to show that $G = HN_G(P)$

I am stumped on this question. Does anyone have some helpful hints or a solution to this question? Thanks! Let $G$ be a group of finite order. Let $H$ be a normal subgroup of $G.$ Let $P$ be a ...
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0answers
29 views

Do involutions suffice to find reflected vectors in a reflection group representation?

Consider a reflection group $W$ acting by isometries on a Euclidean space $V$. I want to understand the union of $(-1)$-eigenspaces for this action, the set $$\{v \in V : \exists w \in W\ (w\cdot v = ...
4
votes
2answers
112 views

Normalizer and centralizer of abelian subgroups of a group are equal [closed]

I have a question: Let $G$ be a finite group. If for each abelian subgroup $H$ of $G$ the centralizer and the normalizer of $H$ are equal, that is, $C_G(H)=N_G(H)$, prove that $G$ is abelian ...
10
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3answers
88 views

Group of order $224$

Any one can give a hint to prove this? Every group of order $224$ have a subgroup of order $28$.
2
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0answers
23 views

Is $B(w_1 w_2)B = (Bw_1B)(Bw_2B)$?

Let $B$ be a Borel subgroup of $GL_n$ and $W$ the Weyl group of $GL_n$. Let $w_1, w_2 \in W$. Is $B(w_1 w_2)B = (Bw_1B)(Bw_2B)$? If this is true. How to prove it? Thank you very much. Edit: If $l(w_1 ...
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1answer
37 views

Reflections - Understanding

I have been asked Does the set of all reflections in the plane form a group? Explain. I thought that the answer was that it wouldn't as if I reflected the parabola $y=x^2$ through the y axis, I ...
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0answers
30 views

Prove that $f(j)=j$ $\forall f \in $ Aut(G) where G is non-abelian & simple [duplicate]

Let $f$ be an element of $\operatorname{Aut}(\operatorname{Aut} G)$ s.t $f(r)$ is equal to $r$ for all $r \in \operatorname{Inn}(G)$. Prove that $f(j)$ is equal to $j$, $\forall ...
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2answers
64 views

$ o(ab)= o(ba)$, What about when $ab$ has infinite order?

I have been able to prove this when I say simply assume $o(ab)= n$, but I have been pondering the case of where $o(ab)$ is infinite? Is this something that needs to be considered? And if not, why? ...
2
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1answer
61 views

Elements of $\operatorname{Aut}(\operatorname{Aut}(G))$ acting as an identity on $\operatorname{Inn}(G)$, for $G$ a nonabelian, simple group

Let $f$ be an element of $\operatorname{Aut}(\operatorname{Aut} G)$ s.t $f(r)$ is equal to $r$ for all $r \in \operatorname{Inn}(G)$. Prove that $f(j)$ is equal to $j$, $\forall ...
1
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1answer
65 views

Elements of $\operatorname{Aut}(\operatorname{Aut}(G))$ acting as an identity on $\operatorname{Inn}(G)$

Let an element $f$ of $\operatorname{Aut}(\operatorname{Aut}(G))$ acts as an identity on $\operatorname{Inn}(G)$ then does it act as an identity on $\operatorname{Aut}(G)$? I have taken an element ...
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0answers
75 views

Is group in which every $a$ satisfies $a^3=e$ abelian? [duplicate]

I know that any group in which every $a$ satisfies $a^2=e$ is abelian. How about if $a^3=e$ for every $a$?
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0answers
34 views

$F/H^{\prime}$ is a torsion free group [closed]

Let $F$ be a free group and $H$ be a normal subgroup of $F$. I want to show that $F/H^{\prime}$ is a torsion free group where $H^{\prime}=[H,H]$.
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0answers
24 views

Is it possible to explicitly demonstrate complex number structures arises naturally from intersecting symplectic groups and orthogonal groups?

Is it possible to explicitly demonstrate complex number structures arises naturally from intersecting symplectic groups $Sp(2n, \mathbb{R})$ and orthogonal groups $O(n)$? My question may not be ...
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0answers
58 views

A sequence of subsets of $\Bbb Z$ not containing nontrivial subgroups

Is there a sequence $(A_n)$ of subsets of $\Bbb Z$ such that always $\{a-b\mid a,b\in A_{n+1}\}$ is a proper subset of $A_n$ and no $A_n$ contains an infinite subgroup of $(\Bbb Z,+)$?
5
votes
2answers
58 views

Subgroups of order $5$ in Icosahedral group

Let $G$ denote the orientation preserving isometries of Icosahedron. I want to show the following using group-theoretic notions: Let $N\leq G$ be a subgroup with order $5.$ Show that it is a ...
2
votes
1answer
60 views

Element of Infinite Order of Finitely Presented Group

Let $G$ be finitely presented group with $n$ generators and $r$ relations where $n>r$. I want to show that $G$ has an element with infinite order. My attempt: Assume that $F_n$ is free group with ...
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1answer
48 views

Kernel of homomorphism from $\mathbb (Q,+)$ to finite group G

If $\phi$ is a homomorphism from $\mathbb Q$ to a finite group $G$, the prove that $$\phi(q) = e_g\forall q\in\mathbb Q \text{ where } e_g \text{ is identity}$$
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1answer
25 views

Subgroups of a group of order 56

I need a hint for the next problem: We have a group $G$ of order 56. Up to isomorphism, there is an unique group of order 56 which doesn't contain normal Sylow $7$-subgroups and the Sylow $2$-subgroup ...
2
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4answers
77 views

If every element of $H$ and $G/H$ is a square , then to prove that so is every element of $G$

Let $H$ be a subgroup of an abelian group $G$ such that every element of $H$ can be written as $b^2 , b \in G$ and similarly for $G/H$ , then how to prove that every element of $G$ can also be written ...
0
votes
2answers
36 views

$H:=\{ g^2 : g \in G \}$ is a subgroup of $G$ $\implies $ $H$ is normal in $G$

Let $G$ be a group . If $H :=\{ g^2 : g \in G\}$ is a subgroup of $G$ , the how to prove that $H$ is normal in $G$ ?
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0answers
69 views

Let $G$ be a group of order 30. Show that $G$ has a subgroup of order 15.

Let $G$ be a group of order 30. Show that $G$ has a subgroup of order 15. Now it has a Sylow 3 & a Sylow 5 subgroup. Next what?
1
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1answer
46 views

$G$ is non-abelian and satisfies $(ab)^2=(ba)^2 , \forall a,b \in G$

It is known that if a finite group of odd order satisfies $(ab)^2=(ba)^2 , \forall a,b \in G$ , then $G$ is abelian . I am looking for examples where (i) $G$ is infinite , non-abelian and satisfies ...
9
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3answers
123 views

Number of elements of order $2$ in $S_n$

How many elements of order $2$ are there in $S_n$? Using combinatorics I arrived at this: For $n$ even ($n=2k$) there are ${n\choose2}+{n\choose 2}{n-2\choose 2}\dfrac{1}{2!}+{n\choose 2} ...
1
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1answer
40 views

A direct proof of $\binom{m\,p^k-1}{p^k-1}\equiv1~(\text{mod $p$})$?

In Nathan Jacobson's "Basic algebra I" the exercises 1.13.11-14 prove the following extension of (a part of) the Sylow's second theorem: If $p$ is a prime and $p^k\bigm||G|$, then the number of ...
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1answer
40 views

Proof of Cauchy's Lemma in the case that G is abelian

I want to prove Cauchy's Lemma for abelian groups: If $G$ is abelian and there exists a prime such that $p$ divides the order of $G$, then there exists a $g \in G$ such that $p=\mathrm{ord}(g)$ I am ...
3
votes
1answer
33 views

Existence of proper I.C.C. subgroup

A countable discrete group $G$ is called I.C.C.(infinite conjugacy class) if for any $e\neq g\in G$, $\#\{sgs^{-1}\mid s\in G\}=\infty$. My question is: Is it possible for a group $G$ to be ...
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1answer
25 views

Frobenius dihedral groups

How can we show by a direct group-theoretic proof that the dihedral group $D_{2n}$ is a Frobenius group iff $n$ is an odd number?
2
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1answer
54 views

Characteristically simple subgroup

Let $G$ be a finite group and $H$ be a minimal normal subgroup of $G$. How can we show that $H$ is a characteristically simple subgroup of $G$?
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0answers
23 views

finite simple group of Lie type

let $G$ be a finite simple group of Lie type. would someone please help me! I want to know, is the following expression true or not? " There exists a simple linear algebraic $\mathcal{G}$ over an ...
2
votes
1answer
37 views

List of groups with specific divisors

I want to find list of finite simple nonabelian groups which their orders divisor lies in specific set of primes for example the set $\{2,3,5,7,11\}$. Is there a method to do this? Would Zsigmondy's ...
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3answers
127 views

What is so special about $a*b^{ -1}$ equivalence?

This equivalent is used often in group theory. For example, using this equivalnce you prove Lagranges theroem and also this equivalence gives you the cosets and other things. This equivalence also ...
2
votes
1answer
63 views

A homomorphism of group $G$ to be an automorphism

Let $G$ be a finite group such that $G=\langle x_{1},...,x_{t}| r_{1}=r'_{1},...,r_{k}=r'_{k}\rangle$. Now we define the homomorphism $\alpha$ of $G$ given by $\alpha({x_{i}})=y_{i}$ for any $i$ such ...
3
votes
1answer
48 views

Minimal normal subgroup in a finite group

Let $G$ be a finite group and $G=HK$ such that $H<G$. How can we show that if $K$ is an abelian minimal normal subgroup of $G$, then $H$ is a maximal subgroup of $G$ and $H\cap K=1$?
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1answer
48 views

Does $G$ is nilpotent imply so is $G/Z(G)$?

If $G$ is nilpotent then is $G/Z(G)$ also nilpotent? If so, how can I prove it? I know the definition of nilpotent group that the upper central series of $G$ goes to $G$ in the finite length.
5
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0answers
40 views

How to “see at a glance” the solution to the exercise “Show that $\langle (1,2,3… n),(1,2,3… m)\rangle$ contains a 3 cycle, if $1 < n < m$”?

I tried the commutator of the generators and it worked, but I had no real justification for making that computation. Is there a perspective from which trying the commutator is the "obvious" thing to ...
3
votes
2answers
50 views

Group theory exercise - verification?

I'm self-studying abstract algebra, and this is the first non-trivial group theory exercise I've done. Although it's a well-known result, I'd like to make sure it is correct as it took a good few ...