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

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45 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 ...
1
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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 ...
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0answers
41 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 $r=90^{\circ}$ ...
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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 ...
1
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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 ...
4
votes
2answers
2k views

A group of order $p^2q^2$ is never simple

Let $p,q$ be primes and let $G$ be a group of order $p^2q^2$, what's the best way to show $G$ is non-simple? I know it suffices to show that one of the sylow-p or sylow-q subgroup of $G$ is normal, ...
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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
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 ...
3
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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 $1$. ...
2
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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 ...
4
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1answer
96 views

Let $G$ denote an arbitrary group. Prove: The center of any group $G$ is a normal subgroup of $G$

Let $G$ denote an arbitrary group. Prove: The center of any group $G$ is a normal subgroup of $G$ Let $G$ be a group and $C$ the center, i.e., for any $a \in C$ and any $x \in G$, $xa=ax$. So, ...
1
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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 $\mathbb{Z}_n*\mathbb{Z}_m$ ...
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, & ...
1
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1answer
83 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, ...
5
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4answers
2k views

Homomorphism between cyclic groups

I have some confusion in relation to the following question. Let $\langle x\rangle = G$, $\langle y\rangle = H$ be finite cyclic groups of order $n$ and $m$ respectively. Let $f:G \mapsto H$ ...
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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 ...
0
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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 $x \in X$, ...
0
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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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1answer
43 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 $(\Bbb Z, +)$?
13
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3answers
2k views

No simple group of order $300$

So I've been trying to prove that there's no simple group of order $300$. This is what I did and I was wondering if it was enough. $|G|=2^2 \cdot 3 \cdot 5^2$. Suppose $G$ is simple. Then there ...
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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
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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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?
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0answers
16 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 ...
2
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1answer
49 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
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
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 ...
2
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1answer
41 views

Isomorphism form $\mathbb{C}[G]$ to $\prod_{i=1}^h M_{n_i}(\mathbb{C})$.

What I want to ask is the proof of the Proposition 10. in "Linear Representations of Finite Groups" by Jean-Pierre Serre. Let $\rho_i : G \rightarrow GL(W_i)$ be the distinct irreducible ...
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1answer
48 views

Defining a ring homomorphism in proving $|C| \cdot \frac{\chi(C)}{\dim V} $

Lemma irreducible representation $\rho: G \rightarrow GL(V)$, $C$ conjugacy class, then $$|C| \cdot \frac{\chi(C)}{\dim V} $$ is an algebraic integer. In the start of this proof we have: ...
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3answers
51 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 ...
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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\\ ...
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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: $\displaystyle \frac{\chi(C)}{\chi(e)}$ ...
5
votes
1answer
92 views

How to eliminate other possibilities of $p$? [duplicate]

Let $G$ be a finite group, $p$ be the smallest prime divisor of $|G|$, and $x \in G $ an element of order p. Suppose $h \in G $ is such that $hxh ^{−1} = x ^{10} $. Show that $p=3$ My try: Since ...
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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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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
38 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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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
44 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 ...
20
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1answer
538 views
4
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2answers
51 views

If $K$ and $H$ are subgroups of $G$ and $H \triangleleft K$ then $K \subseteq N(H)$.

We can easily prove the truth of this statement. My question is that why do we not simply say $K=N(H)$ $?$ I'd be really grateful for an elaboration on this.
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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 ...
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 ...
1
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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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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 ...