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

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19 views

Prove that there exits an automorphism from $G$ to $G$ when dim G=infinite

Suppose $G$ is a vector space over $\mathbb Z_2$ . The problem is to prove that there exits an automorphism from $G$ to $G$ Now $G$ has a basis say $\{b_1,b_2,...,b_n\}$.Then any $g\in G$ can be ...
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2answers
35 views

Prove that $G$ is a vector space over $\mathbb Z_2$

Suppose $G$ is an abelian group such that all non-identity elements in $G$ has order $2$. Prove that $G$ is a vector space over $\mathbb Z_2$ Since $G$ is an abelian group only thing to show is to ...
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0answers
19 views

Is trace of regular representation in Lie group a delta function? [duplicate]

My major is physics. I need to use some tools in group theory, but I am really confused by the trace in compact infinite groups. The following is my question: In discrete group theory, the ...
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25 views

generators for group for the given finite field? [closed]

Question is "Multiplicative group of nonzero elements of a finite field is cyclic. Illustrate this by finding a generator for this group for the given finite field" a) Z7 b)Z11 c)Z17 I don't even ...
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1answer
58 views

Is trace of regular representation in Lie group a delta function?

My major is physics. I need to use some tools in group theory, but I am really confused by the trace in compact infinite groups. The following is my question: In discrete group theory, the ...
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0answers
8 views

Commutation Relation - Generators of Semisimple Lie Group

The following is stated in a book I picked up, Group Theory for High-Energy Physicists - M. Saleem, M. Rafique. Consider an $r$-parameter semisimple Lie group of rank $\ell$. It has a set of ...
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25 views

Geometric understanding of why $D_1\cong \mathbb{Z}_2$

From what I understand, $P_2$ is the line joining $(-1,0)$ to $(1,0)$ and then $P_1$ is the point $(1,0)$. This is due to defining $P_n$ (the regular polygon with # of points $n$) to be formed by the ...
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2answers
86 views

Conjugacy classes of the group $G=\langle a,b\mid a^6=1,b^2=a^3,bab^{-1}=a^{-1} \rangle$

Consider the group $G=\langle a,b\mid a^6=1,b^2=a^3,bab^{-1}=a^{-1} \rangle$. This group does not appear to be easy to work with! Does anyone know what this group is called? I am trying to find its ...
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2answers
47 views

What would the notation G/H mean in terms of groups and subgroups?

What would G/H mean in terms of subgroups? Would it most likely mean The compliment group of H in G?
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3answers
44 views

An example of a group that that has an order of M that is abelian?

Theory: If G is a finite abelian group, p is prime and p divides the order of G then G has an element of order p. Can anyone think of a counter example for a number n that is not prime, divides the ...
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65 views

If a certain group action fixes every element $x$ such that $x^4=1$, then the action is trivial

This is a question from chapter $4D$ of Isaacs' Finite Group Theory. Let $A$ act via automorphisms on $G$, where $G$ is a $2$-group and $A$ has odd order. Show that if $A$ fixes every element $x$ in ...
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1answer
49 views

Using generators to write representations

Let $G=D_{12}=\{a,b\mid a^6=b^2=1, bab=a^{-1}\}$. Also let $A=\begin{pmatrix} e^{i\pi/3} & 0 \\ 0 & e^{-i\pi/3} \end{pmatrix}$ and $B=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$. ...
2
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1answer
45 views

Show that $\det \rho(g)=1$ for elements g of odd order and $\det \rho(g)=-1$ for elements g of even order.

$G$ is a finite group. $\rho$ is a representation of $G$, then $\rho \mapsto \rho(g)$ is a $1$-dimensional representation. Show that if $\det \rho(g)=-1$ for some $g \in G$ then $G$ has a ...
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1answer
38 views

Only one cancellation law? Then $G$ may not be a group…

Suppose that the following result is known: "Let $G$ be a finite set, closed with respect to an associative product and that both of the cancellation laws are valid. Then $G$ is a group with ...
2
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1answer
24 views

Prove that the inner product is the number of orbits of $G$ on $X \times Y$.

Let $X,Y$ be $G$-sets and $\mathbb{C}[X], \mathbb{C}[Y]$ the corresponding permutation representations. Prove that the inner product is the number of orbits of $G$ on $X \times Y$. Ive tried: ...
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5answers
34 views

Product of subgroups which is not a group - infinite case

It is well known that a product of subgroups might not be a subgroup. I give an example here. Would you have an example of an infinite group having two infinite subgroups whose product is not a ...
2
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1answer
52 views

Number Theory : Is a complete residue system modulo $n$ a group?

I was working my way through some basic number theory problems, when in the chapter on "Introduction to Group Theory," I came across the following: Show that for every positive integer $n$, the ...
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0answers
29 views

Is there a general method to calculate the generators of the subgroups of $\textrm{GL}(n,F)$?

I know this might be a very bad/broad question, but after going through a few practice problems for finding linearly independent generators for some of the easier subgroups of $\textrm{GL}(n,F)$ ...
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0answers
31 views

Direct limit of quotient groups

For a subgroup $H$ of $G$, we denote $\langle H\rangle $ be the smallest normal subgroup of $G$ containing $H$. (That is, the normal closure of $H$ in $G$.) Suppose that ...
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1answer
28 views

What is the process behind finding a Cayley permutation representation.

For example, let's find the Cayley permutation representation of $\mathcal D_3$ in $S_6$. $\mathcal D_3 = \left<r,s \mid r^3=s^2=1, rs=sr^{-1}\right>$. Write, \begin{pmatrix} 1 & 2 & ...
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3answers
92 views

Let $G$ be a finite simple group. Suppose that $A, B < G$, $G = AB$ and $A$ is an Abelian group. Is it true that $A \cap B=1 $?

Let $G$ be a finite simple group. Suppose that $A$ and $B$ are proper subgroups of $G$, $ G = AB$ and $A$ is an Abelian group. Is it true that $A \cap B=1 $ ? I checked it with some examples and it ...
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2answers
126 views

How to show that the union of an infinite sequence of subgroups is a subgroup?

I'm self-studying from Algebra: Chapter 0 by Aluffi. I was working on Exercise 6.6, and I can do the first part (which I've included for context), but I'm having trouble with the second part. Here is ...
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2answers
57 views

How do I find the permutation with the highest order in a symmetric group?

My professor gives this text, but I don't understand what it's saying, could someone explain it to me? Let $M(n)$ denote the largest order of an element in $S_n$. By Theorem 1 $M(n)$ is the largest ...
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25 views

If $P = \langle y \rangle X$ for some element $y$. Let $P > 2 $ then $1 \neq t \in N \cap Z(P)$, [closed]

Let $P$ be a $p$-group and let $N$ be a nontrivial, elementary abelian normal subgroup of $P$ which has a complement $X$ in $P$. If $P = \langle y \rangle X$ for some element $y$. Let $P > 2 $ ...
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2answers
49 views

I have a question about group G which satisfies Inn(G) char Aut(G) and Z(G)={1}.

Let $G$ be a group which satisfies $Z(G)=\{1\}$ and $Inn(G) \space \mathbb{char} \space Aut(G)$; then every automorphism of $A=Aut(G)$ is an inner automorphism. ($H \space \mathbb{char} \space G$ ...
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2answers
41 views

For $n \ge 3$, every subgroup of $A_n$ with index $n$, is isomorphic to $A_{n-1}$

For $n \ge 3$, every subgroup of $A_n$ with index $n$, is isomorphic to $A_{n-1}$. Any idea to solving it?
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1answer
31 views

generating system of the kernel of a module-transformation

Let $G ≠ 1$ be a group and A a commutative ring. Now, the group ring $A[G]$ is naturally an A-module. Next, let's consider the transformation: $$\phi: A[G] \to A, \sum_{g \in G} a_gg \mapsto \sum_{g ...
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50 views

Show that $\rho$ must be 2-dimensional

Let $G=D_8=\langle g,h |g^4=h^2=1, hgh=g^{-1} \rangle$. One can show that $G$ has $4$ $1$-dimensional representations. From first principles (no character theory). Suppose $\rho$ is an ...
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2answers
54 views

How to read a cycle graph?

As an important tool for visualizing some small finite groups it is useful to know how read such graph, and with time trying to make sketch of them by my own. I would like to know, for a start, how ...
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1answer
84 views

When is $G\cong\operatorname{End}(G)$?

$\newcommand\End{\operatorname{End}}$Let $G$ be an Abelian group. Are there sufficient conditions for the existence of an isomorphism $G\cong\End(G)$, where $\End(G)$ is considered a group under ...
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0answers
56 views

Prove that $H$ is a subgroup of $G$ [closed]

Either prove that $H$ is a subgroup of $G$ or determine why it is not. $(a) \ G = \mathbb R$ and $H = \mathbb R^+$ $(b) G = \mathcal D_6$ and $H = \{e,r^2,r^4, m_1, m_3, m_5\}$ Where $r=90^{\circ}$ ...
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1answer
57 views

Need of proof of associative property of symmetric group

I'm a novice in learning group theory.Recently I'm learning about symmetric group. I understood the closure property,identity property and inverse property of symmetric group but I don't understand ...
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1answer
30 views

How it works as cyclic group, somebody do explain it.

For a geometrical realisation of a cyclic group, let $S$ be the circle, in the plane, of radius $1$ , and let $\rho_n$ be a rotation through an angle of $\frac{2 \pi}{n}$. Then $\rho_n \in A(S)$ and ...
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1answer
54 views

Why the orbit is of dimension $12$?

Let $SL_3$ acts on the variety consisting of all nilpotent $3$ by $3$ matrices over $\mathbb{C}$ by conjugation. Let $S_p$ be the orbit of the matrix $$ a=\left( \begin{matrix} 0 & 1 & 0 \\ 0 ...
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2answers
44 views

If $F$ is a free group with basis $X$ and $N = \langle \{ g^2 : g \in F\} \rangle$, why is $F/N$ elementary abelian of rank $|X|$?

This seems to be a standard trick - take the subgroup $N$ generated by all squares of elements in a group $G$. Then $N$ is normal, since the conjugate of a square is a square, and $G/N$ is abelian ...
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1answer
73 views

Let $G$ be a finite group and let $P$ be a Sylow p-subgroup with $N_G(P)=P$. Prove that $P$ is not contained in any proper normal subgroup of $G$.

Let $G$ be a finite group and let $P$ be a Sylow p-subgroup with $N_G(P)=P$. Prove that $P$ is not contained in any proper normal subgroup of $G$. We can suppose $P \subseteq K\lhd G$. Since $P$ is ...
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2answers
74 views

Why do we think of group compositions as multiplication?

This has bothered me for some time: The composition in a group is usually denoted $xy$ or $x\cdot y$. Powers (note the word) are denoted by $x^n$, inverses by $x^{-1}$, and the neutral element by $1$. ...
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1answer
65 views

$G$ is a finite group,with eight Sylow 7 subgroups. Show that there exists a normal subgroup $N$ of $G$ s.t. index $[G:N]$ divisible by 56, not by 49.

Let $G$ be a finite group which has exactly eight Sylow 7 subgroups. Show that there exists a normal subgroup $N$ of $G$ such that the index $[G:N]$ is divisible by 56 but not by 49. I will start by ...
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1answer
24 views

Are free products of finite cyclic groups perfect?

I read that $\text{PSL}(2,\mathbb{Z})$ is isomorphic to $\mathbb{Z}_2*\mathbb{Z}_3$, which is a perfect group. Then, in general, for natural numbers $n$ and $m$, when is $\mathbb{Z}_n*\mathbb{Z}_m$ ...
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1answer
43 views

Finding a group that is not monomial

Definition. A group is called monomial if every representation of $G$ is induced from 1-dimensional representations of some subgroup of $G$. Question Give an example of a group that is not monomial. ...
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2answers
34 views

Finding the factor of the derived subgroup of non-abelian group of order 12

Consider the group $G=\langle a,b\mid a^6=1,b^2=a^3,bab^{-1}=a^{-1} \rangle$. The derived subgroup is $G'=\{e,a^2,a^4\}$ I believe. So I am trying the find $G/G'$. Now I know that $|G/G'|=4$ so it ...
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0answers
57 views

Number of group homomorphisms between two finite groups

I am confused between the Answer of this Question 1 and the Answer of this Question 2. In answer of the 1st question groups should must be Abelian whether in the answer of 2nd question there are no ...
2
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0answers
32 views

Proof of the Frobenius Schur indicator

I am trying to prove the Frobenius-Schur indicator for $\chi$ irreducible character. \begin{equation} i_{\chi} = \begin{cases} 0, & \text{if $\chi$ is not real valued} \\ \pm1, & ...
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1answer
39 views

What it means SO* (2N)?

I'm puzzled about the $"*"$ in the following notation for Lie groups: $SO^* (2N)$ or $SU^* (2N)$. I don't understand what is the meaning of this notation. It is introduced for example in Gilmore ...
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1answer
79 views

What is the exponent of a group?

I don't really understand the definition: The exponent of a group G is the smallest natural number x such that for all $g \in G,g^x = e$. It seems like it's saying, for EVERY element of the group, ...
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21 views

Sylow p,q,r-Subgroups

I am quite new to group theory, so I am trying to get my head around Sylow's Theorems and other stuff....I got an exercice here and I am not sure how to go on with the proofs. We have a group G of ...
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2answers
27 views

How can a subgroup have multiple cosets?

I am currently reading An Introduction To The Theory Of Groups, by Joseph Rotman, and in a section describing cosets, there is an exercise question as follows; Let $H$ be and subgroup of $G$ having ...
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0answers
21 views

What's about the order of this group? [closed]

I know for a nonempty set $S$ containing $n$ elements the group $A(S)$ contains n! elements. But if $S= \{-1, 0, 1\}$ and if $A(S)$ be the group under addition, then here $o(A(S))$ is not $3!$ . So ...
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12 views

Components and Centralisers of Involution. A wrong argument?

A component $K$ in a finite $G$ is a subnormal subgroup which is quasisimple, i.e. perfect and $K/Z(K)$ is simple. Obviously, when $K$ is a component of $G$ and $U\le G$, then $K$ is also a component ...
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0answers
32 views

Linear groups and isomorphisms

If two linear groups(subgroups of $\text {GL}(n,k)$ over some field $k$) $G(t)$ and $H(t)$ over $F(t)$, a transcendental extension of a field $F$, are isomorphic, then for each $f\in F$, are $G(f)$ ...