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

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3
votes
1answer
40 views

Polycyclic groups and Group extension

Suppose we have a SES $1\to N\to G\to G/N\to 1$, and assume that $G/N$ is polycyclic. What condition on $N$ will ensure that $G$ is polycyclic? THanks.
2
votes
1answer
39 views

Characterizing groups with linear subgroups

Is there a simple characterization for a group $G$ satisfying $$(\forall H,K\le G)(H\subseteq K\text{ or } K\subseteq H)$$
3
votes
0answers
35 views

When can we find coset representatives that generate the original group?

Let $G$ denote a group and consider a subgroup $H \subseteq G$. Then the left cosets of $H$ form a partitioning of $G$ (irrespective of whether or not $H$ is normal). Call this partitioning $\Pi_H$. ...
1
vote
0answers
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 $G$. ...
1
vote
1answer
29 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.
6
votes
4answers
93 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 ...
0
votes
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
6
votes
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
votes
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
votes
3answers
131 views

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

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
votes
1answer
73 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
votes
1answer
44 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
votes
0answers
38 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 ...
1
vote
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 ...
5
votes
2answers
104 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 ...
0
votes
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 ...
1
vote
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
vote
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
votes
0answers
42 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
vote
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
votes
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
votes
2answers
72 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
104 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 ...
1
vote
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
votes
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
32 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
102 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
votes
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 ...
0
votes
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 ...
1
vote
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
114 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
votes
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
votes
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 ...
1
vote
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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votes
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 ...
1
vote
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
votes
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
vote
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 ...
1
vote
0answers
77 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 $a$?
2
votes
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 $H^{\prime}=[H,H]$.
0
votes
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 ...
1
vote
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 ...
1
vote
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 $$\phi(q) = e_g\forall q\in\mathbb Q \text{ where } e_g \text{ is identity}$$
1
vote
1answer
26 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
votes
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 $G$ ?
0
votes
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?