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

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31 views

Both elements are identity [duplicate]

In a group $G$, suppose there are elements $a,b\in G$ satisfying $$ a^{-1}b^2a=b^3\quad \text{ and }\quad b^{-1}a^2b=a^3$$ How to show that $a=b=\rm e$. Where $\rm e$ is the identity element of $G$. ...
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1answer
27 views

Order of a group and cyclic group theory connection

Just stuck on a problem. If group $G$ of order $6$ contains an element of order $6$, then prove that $G$ is a cyclic group of order $6$. Any hint will be appreciated.
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4answers
89 views

An infinite group $G$ and $\forall x\in G, x^n=e$

Let $G$ be an infinite group and $n\in \mathbb N$. If for any infinite subset $A$ of $G$ there is $a\in A$ such that $$a^n=e,~~~~(e=e_G)$$ then prove that for every element $x\in G$ we have ...
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1answer
37 views

There is no group whose quotient by the center is isomorphic to the quaternion group [duplicate]

I have to prove that there is no group $G$ such that $G/Z(G)$ is isomorphic to $Q_8$. Anytone can give me an idea to begin? thanks
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2answers
70 views

Groups, inverse Galois problem and transcendence degrees

This is a curiosity of mine. I suspect there might be a trivial answer, but if there is none, this problem will probably haunt me for a long time... The question is as follows : Given a group $G$, ...
2
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1answer
53 views

$GL_2(R)$ and $\mathbb{A}_4$

So, i have to prove that $\mathbb{A}_4$ is not isomorphic to a subgroup $G$ of $GL_2(R)$. Here is a "hint" (are previuos part of the same problem that i was thinking, so i supose should help): If ...
2
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3answers
129 views

Show that $\mathbb Z[x]$ and $\mathbb Q_{>0}$ are isomorphic [on hold]

Let $(\mathbb{Z}[x],+)$ be the additive group of all polynomials with integer coefficients and $ (\mathbb{Q}_{>0},*)$ the multiplicative group of all positive rationals. (Please) Show (me) these ...
6
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1answer
72 views

Field extension $\mathbb Q(f)/\mathbb Q$ and its Galois group

Let $E/\mathbb{Q}$ and $F/E$ be finite extensions of fields, let $u$ be an element of $Aut(F/E)$, and let $f$ be an element of $F$. Suppose that (i) $[F:E]=3$, (ii) $F=\mathbb{Q}(f)$ and ...
2
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1answer
40 views

The Galois group of a polynomial

I was just looking for some clarification regarding the definition of the Galois group of a polynomial $f(x)$. So, if I remember correctly, this is defined as the Galois group of a splitting field of ...
4
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37 views

Figuring out automorphism groups

I was wondering what tactics people usually use to figure out automorphism groups. Let's start with small finite groups. For example, I'm trying to figure out $\mathrm{Aut}(V_4)$. My thought process ...
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1answer
24 views

The number maximal subgroups of a 2-generated group

Let $G$ be a 2-generated group. Then prove that the number subgroups of index 2 is at most 3. By Hint i think we have at most 3 cases: Let $G=\langle a,b\rangle$ and $C_{2}=\langle x\rangle$. Then ...
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2answers
99 views

About Commutators in Subgroups

Let $G$ be a group and $H$ a subgroup of $G$. Is clear that if $x$ and $y$ are elements in $H$ then $[x,y] = x^{-1}y^{-1}xy \in H$. But, is true that, if $1 \neq [x,y] \in H$, then $x$ and $y$ are ...
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2answers
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
30 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 $C_{G}(x)=\{g \in G: xg=gx\}$ ...
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0answers
48 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 ...
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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 ...
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1answer
50 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 $H$. ...
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3answers
62 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 ...
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2answers
70 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 ...
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2answers
94 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, ...
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1answer
14 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
23 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 ...
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1answer
81 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 ...
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2answers
28 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 $Q\subset X$ ...
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1answer
100 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 ...
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1answer
46 views

If $f(N) = M $, $\ker f < N$, then the preimage of $M$ is contained in $N$? [closed]

This is from "Algebra" by Hungerford (page 44, proof from corollary 5.8) Let $f:G \rightarrow H$ be a homomorphism, $N$ be a normal subgroup of $G$, and $M$ be a normal subgroup of $H$. Show that if ...
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2answers
68 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
28 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
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2answers
103 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 ...
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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 $28$.
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43 views

Suggestions for choice of abstract algebra project [closed]

I want to know which of these topics have more materials and is interesting to write my project on... Group action as an extension of group multiplication or Wreath products of cyclic groups
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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? ...
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1answer
60 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 ...
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1answer
56 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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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
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2answers
57 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
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1answer
59 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
47 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 ...
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4answers
70 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 ...
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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
63 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 ...