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

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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
110 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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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? ...
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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
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 ...
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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 ...
1
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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 $(\Bbb Z,+)$?
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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
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 ...
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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
67 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 ...
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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} ...
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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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votes
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?
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1answer
53 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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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
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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 ...
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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.
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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 ...
2
votes
1answer
50 views

About a nested sequence of subsets of integers

Let $(H_n)$ be a sequence of nonempty subsets of $\Bbb Z$ such that always $\{a-b\mid a,b\in H_{n+1}\}\subsetneqq H_n$. Can we deduce that there is some $n$ such that $\{a-b\mid a,b\in H_{n}\} = ...
3
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1answer
47 views

Find $o(b)$ if $aba^{-1}=b^2$ and given that $a^5=e$

If in a group $G$, $a^5=e$, $aba^{-1}=b^2$ for some $a,b\in{G}$. Find $o(b)$. I wrote $aba^{-1}=b^2$ as $ab=b^2a$. Then $(ab)^5=(b^2a)^5$ but then I am stucked up.
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0answers
39 views

A question from Topics in algebra-IN Herstein. [duplicate]

Prove that if a abelian group has elements of order $m$ and $n$ then it has a subgroup of order equals to $lcm[m,n]$. I am new to group theory so please explain....
1
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1answer
38 views

On group that has no non-abelian subgroup of order 6

Do there exits a group $G$ such that $G$ has not a non-abelian subgroup of order 6, but $Inn(G)$ is isomorphic to a non abelian group of order 6? Thank you We know that $\dfrac{G}{Z(G)}\cong ...
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1answer
47 views

Let $p$ be a prime number and let $Z=\{z\in \mathbb{C}\: z^{p^{n}}=1$ for some $n \in N\}$.

Let $p$ be a prime number and let $Z=\{z\in \mathbb{C}: z^{p^{n}}=1$ for some $n \in\Bbb N\}$. (a) Show that every proper subgroup of $Z$ is of the form $H_{k}$ for some $k$, where $H_k=\{z\in C: ...
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votes
1answer
52 views

Is there any efficient algorithm for finding subgroups of a given finite group?

I am implementing an algorithm which finds every subgroup of given group. Here's my algorithm. Let $G$ be a group of order $n$ with elements $g_1,\cdots,g_n$. Then I consider each $\langle ...
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vote
1answer
35 views

Show that $|5|=2^{n-2}$ when $n \ge 3$ in $U(2^n)$

Show that $|5|=2^{n-2}$ when $n \ge 3$ in $U(2^n)$. I wrote ...
2
votes
2answers
58 views

Cardinality of Centralizer of some element

Let $G$ be a finite group. How can we show that $|G/G^{'}|\leq |C_G(x)|$ for all elements $x\in G$?
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0answers
20 views

How to find out exactly invariant factor decomposition of finitely generated abelian groups

Suppose that we defined some finitely generated abelian group $G$. Now how does one find invariant or primary decomposition of $G$? We know that decomposition exists, how do we exactly state ...
2
votes
1answer
29 views

Extension theory and automorphism extension

My question is motivated by the following two posts On finite 2-groups that whose center is not cyclic and Automorphisms of group extensions Question: Assume that $A,B,C$ are there algebraic ...
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2answers
172 views

How can I find $\mathbb Z_4$ as an extension of $\mathbb Z_2$ by $\mathbb Z_2$?

Let $H=\{1,h\}$ and $A=\{0,a\}$ be groups, and $\pi:H\rightarrow \text{Aut}(A)$ be the trivial homomorphism. I have found $FS(H,A,\pi)=\{f_0,f_1\}$ and $IFS(H,A,\pi)={f_0}$ where ...
2
votes
1answer
48 views

Groups with order $p^3$ ($p$ prime) have two non commutative isomorphism classes

I read in an exercise, that a group with $p^3$ ($p$ prime) elements have $2$ non commutative isomorphism classes. Unfortunately there was just this statement without any explanation. We just solved it ...
4
votes
2answers
50 views

A question on terminology (Group Theory)

Is there a standard adjective to describe a finite group $G$ of composite order which possesses, for each (positive) divisor $d$ of $|G|$, a subgroup of order $d$? I would guess "Lagrangian" but I ...
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4answers
49 views

A finite group of order $n$, having a subgroup of order $k$ for each divisor $k$ of $n$, is not simple?

I was asked to prove that, if a finite group $G$ of order $n$ has a subgroup of order $k$ for each divisor $k$ of $n$, then $G$ is not simple. I tried to do this but I could not. Can anyone please ...
0
votes
0answers
30 views

If the automorphism group of a group is cyclic, then the group is commutative [duplicate]

Let $G$ be a group and the $Aut(G)$ group is cyclic $\Rightarrow$ the group $G$ is commutative. I looked at the homomorphism $\varphi : G \rightarrow Aut(G) \ g \mapsto (x \mapsto gxg^{-1})$. Let ...