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

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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, & ...
2
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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
80 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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0answers
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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0answers
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 $G(f)$ ...
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2answers
17 views

Stabilisers of group action open imply the action is continuous

Let $\mu \colon X \times G \longrightarrow X$ be the action of a topological group on a set $X$. We consider $X$ to be a topological space with the discrete topology. Suppose that for all $x \in X$, ...
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1answer
31 views

Finite representations of the Euclidean Group

What are the finite dimensional indecomposable representations of the special Euclidean group in three-dimensions, SE(3)? To clarify, I'm asking about the group $$SE(3) = \left\{ \begin{pmatrix} ...
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0answers
40 views

Prove that $H$ is a subgroup of $G$ or determine why it is not. [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 $r=90^{\circ}$ ...
-1
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1answer
39 views

What is the order of each element of $(\Bbb Z, +) $ [closed]

Given the Group $(\Bbb Z, +)$, what is the order of each element of $(\Bbb Z, +)$?
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1answer
36 views

In the Group $(S_{3},\circ)$, what are the elements of the group $\Big(\big((123)\big), \circ\Big)$?

Given the Group $(S_{3},\circ)$ What are the elements of the group $\Big(\big((123)\big), \circ\Big)$? Also, why does $\big((123)\big)$ have two brackets around it?
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1answer
20 views

In the Group $(G, .)$, where $G=\{1,-1,i,-i\}$, What is $O(1)$, $O(-1)$, $O(i)$ and $O(-i)$?

In the Group $(G, .)$, where $G=\{1,-1,i,-i\}$, What is $O(1)$, $O(-1)$, $O(i)$ and $O(-i)$? My Answer: 1,2,4,4 is that right?
1
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1answer
36 views

Prove that $a^{i}=b^{j}$ if and only if $i\equiv{j}\pmod{n}$ [closed]

Let $(G, *)$ be a Group, $a\in{G}$, and $O(a)=n$. Why is $a^{i}=b^{j}$ if and only if $i\equiv{j}\pmod{n}$?
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0answers
15 views

Tree structure by using integer markers

I'm trying to model a situation in witch a group of entities are organized hierarchically. We say that entity A has privileges over entity B if there a direct hierarchical connection between A and B ...
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1answer
22 views

Size of Dihedral Group with Orbit-Stabilizer

Use the orbit-stabilizer theorem to determine the size of the symmetry group of a regular n-gon. I know that the Dihedral group has order $2n$, but I am having trouble using the orbit-stabilizer ...
1
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1answer
29 views

The action of real special linear group on the complex plane

Let $SL_2(\mathbb{R})\curvearrowright \mathbb{C}\cup\{\infty\}$ where $Az=\frac{az+b}{cz+d}$ Show that if $z=x+iy$ with $y>0$ (has positive imaginary part) then $Az$ does too. Then, considering ...
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1answer
33 views

Conjugacy of Symmetric group [closed]

Let $G=S_n$. Show that $\sigma(x_1x_2...x_k)\sigma^{-1}=(\sigma(x_1)\sigma(x_2)...\sigma(x_k))$ Also, describe the conjugacy classes of $S_5$. I'm having trouble getting started. Any helpful tips or ...
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3answers
49 views

Proof of normal subgroup

The question is: There's a group G, with order pm, where p is a prime number and mcd(p,m) = 1. We suppose that G has an unique p-Sylow subgroup P. Proof that P is a normal subgroup of G. How I ...
6
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2answers
53 views

The minimal group with Fitting length three

Let $G$ be a group with Fitting lengt $3$ i.e $$e< F_1< F_2 < F_3=G$$ and $F(G)=F_1$ and $\bar {F_2}=F(G/F_1)$. If every proper subgroup of $G$ and every non-trivial quatient of $G$ has ...
0
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1answer
26 views

Sufficient Conditions for the Commutator Subgroup to be a Component

A group $K \ne 1$ is quasisimple if $K$ is perfect and $K/Z(K)$ is simple. For every subnormal subgroup $N$ of a quasisimple group $K$ either $$ N \le Z(K) \quad \mbox{ or } \quad N = K. $$ A ...
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1answer
56 views

$GL_3(\mathbb{F}_2)$ is simple

Character table of $GL_3(\mathbb{F}_2)$. \begin{array}{|c|c|c|c|c|c|} \hline & 1 & 2 & 3 & 4 & 7A & 7B \\ \hline \chi_1 & 1 & 1 & 1 & 1 & 1 & 1\\ ...
2
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1answer
48 views

About Cayley graphs on finite fields.

If one is given $n$ vectors of length $n$ $\in \mathbb{F}_{p^k}^n$ for some prime number $p$ and $k \in \mathbb{Z}^+$ then how can one check if they are linearly independent? (the issue is if there ...
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1answer
21 views

In the group $(S_{5}, \circ)$, what are the elements generated by $(12345)$? [closed]

Given the group $(S_{5}, \circ)$, what are the elements generated by $(12345)$?
1
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0answers
25 views

What does it mean for a subgroup to be self-centralising in terms of group extensions

In texts on group theory I read about subgroups $U \le G$ which fulfill the property $$ C_G(U) \le U $$ (this is called self-centralising, for example the Fitting subgroup in solvable groups fulfills ...
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1answer
37 views

Assuming the axiom of choice ,how to prove that every uncountable abelian group must have an uncountable proper subgroup?

Assuming the axiom of choice , how to prove that every uncountable abelian group must have an uncountable proper subgroup ? Related to Does there exist any uncountable group , every proper subgroup ...
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2answers
43 views

Can a cyclic group have more than two generators? [duplicate]

Can a cyclic group have more than two generators? for example the group $\mathrm{Z}$ has two generators $-1$ and $1$, but can a group have more than two generators?
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1answer
34 views

End of the proof of Burnsides $p^aq^b$ Theorem

Theorem. Every group of order $p^aq^b$ ($p,q$ primes $a,b \geq 0$) is soluble. Near the end of the proof of Burnsides $p^aq^b$ Theorem, we want to prove the following If $\rho:G ...
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0answers
34 views

Middle bit of the proof of Burnsides $p^aq^b$ Theorem

Theorem. Every group of order $p^aq^b$ ($p,q$ primes $a,b \geq 0$) is soluble. Part of the proof needed to prove the above theorem is where you prove that: $\displaystyle \frac{\chi(C)}{\chi(e)}$ ...
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1answer
537 views

Does there exist any uncountable group , every proper subgroup of which is countable?

Does there exist an uncountable group , every proper subgroup of which is countable ?
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3answers
54 views

Is the pair $(\{0,4,8,12\},+_{16})$ consist a Group?

Is the pair $(\{0,4,8,12\},+_{16})$ consist a Group? I think $\{0,4,8,12\}$ is written as $\{[0],[4],[8],[12]\}$, where $ [a]=\{x\in{Z}:x=a+n.k,k\in{Z}\} $ I don't understand how this is a group ...
1
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1answer
34 views

Number of orthonormal sets of vectors $\leq$ dim of the vector space

Theorem. If $V,W$ are arbitrary representations of $G$, say $$V=V_1^{a_1}\oplus \dots \oplus V_k^{a_k} $$ $$W=V_1^{b_1}\oplus \dots \oplus V_k^{b_k} $$ Then, $$\langle \chi_V,\chi_W \rangle ...
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1answer
31 views

Characteristic subgroup of an Abelian-by-Finite Group

Let $G$ be a group such that $A$ is a normal Abelian subgroup and $G/A$ is finite. Is always possible to find an Abelian characteristic subgroup $B$ such that $G/B$ is finite too? Factoring by $G^n$ ...
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2answers
38 views

Start of proof of Burnsides $p^aq^b$ Theorem

Theorem. Every group of order $p^aq^b$ ($p,q$ primes $a,b \geq 0$) is soluble. Proof. Enough to prove that no non-abelian simple groups have order $p^aq^b$. [Then break $G$ into simple pieces ...
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0answers
30 views

Understanding definition of soluble

Definition G is soluble if there is a chain of subgroups $$\{e\}=G_n \triangleleft G_{n-1}\triangleleft\dots\triangleleft G_0=G$$ with $G_{i-1}/ G_i$ abelian (or cyclic, or cyclic of prime ...
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2answers
42 views

In an abelian group, prove that $x=a_1a_2\cdots a_n$ implies $x\circ x=e$ [closed]

Let $(G,\circ)$ be a group with elements $a_1,a_2,\cdots, a_n$ and $x=a_1a_2\cdots a_n$. Show that $x\circ x=e$.
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0answers
40 views

If in a finite group $x^3y^3 = (xy)^3$ and $3\nmid|G|$, then $G$ is abelian [duplicate]

Let $G$ be a finite group. For all $x,y \in G$, $x^3y^3 = (xy)^3$. Show that if $3\nmid |G|$, G is necessarily abelian. Tried my hand at it for days.
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1answer
69 views

Infinite dimensional linear groups.

Suppose we have a group of matrices over finite field $\mathbb F_p$ such that in every row and in every column there are only finitely many elements from $\mathbb F_p$. (Also every finite set of ...
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0answers
21 views

How to write su(3) Lie algebra as a sum of two subspaces? [duplicate]

Let K,F⊂su(3) be subspaces, such that K⊕F=su(3), and K has a su(2) structure. How can we show that [K,K]=K (i.e., commutator of any two elements of K gives an element in K), [K,F]=F, and [F,F]=K?
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2answers
59 views

Finding an operation on $G^S$ that yields a group

Problem: Assume $S$ is a nonempty set and $G$ is a group. Let $G^S$ denote the set of all mappings from $S$ to $G$. Find an operation on $G^S$ that will yield a group. Update (full attempted ...
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1answer
40 views

left and right multiplication for Cayley graphs

Correct me if i am wrong but i have written down the following: If $X$ is a finite group, with subset $S$ and corresponding Cayley graph $G$ The edge set for a Cayley graph is defined such that two ...
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1answer
28 views

List all homomorphisms from $\mathbb{Z_2}$ to $\mathbb{Z_4}$

I have only come up with two. That is $f(1)=1, f(1)=0$. Are there any more? If so, how should I go about thinking this problem through, to ensure that I have found all of them?
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0answers
34 views

Generator of group, Computation of discrete logarithm

The prime number $p=67$ is given. Show that $g=2$ is a generator of the group $\mathbb{Z}_p^{\star}$. Compute the discrete logarithm of $y=3$ as for the base $g$ with Shanks-algorithm. Compute the ...
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3answers
40 views

Clear up definition of cayley graph

I have come across two definitions of Cayley graphs, both very similar but one being more general. I have been working with the more general definition which is: A Cayley graph of a group 􏰎$X$ ...
2
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0answers
55 views

Where can I find Wielandt's original proof of Sylow's Theorem?

I have seen several proofs of Sylow's Theorem based on Wielandt's method. Everyone gives credit to Wielandt's proof of Sylow's theorem, but ironically everyone puts their own spin on it. Where can I ...
2
votes
0answers
31 views

How to write $\mathfrak{su}(3)$ Lie algebra as a sum of two subspaces?

Let $K,F\subset\mathfrak{su}(3)$ be subspaces, such that $K \oplus F =\mathfrak{su}(3)$, and $K$ has a $\mathfrak{su}(2)$ structure. How can we show that $[K,K] = K$ (i.e., commutator of any two ...
1
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1answer
54 views

Find $\alpha$ such that the given field is $\mathbb{Q}(\alpha)$ [duplicate]

This question is in regards to separable field extensions. I am to show that this $\alpha$ is in the given field and verify by direct computation that the given generators for the extension of ...
2
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0answers
22 views

QR decomposition of finitely generated matrix groups

Given a finite (or more generally, compact) set of matrices $\{A_i\}\subseteq\mathbb{R}^{n\times n}$, we can generate a subgroup $G$ of $GL(\mathbb{R},n)$. Now, since every matrix has a QR ...