Tagged Questions

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

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0
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13 views

How is number of conjugacy class related to the order of a group?

Let $c(G)$ denote the number of conjugacy classes of a group $G$. I have to show that $$\lim_{|G|\to \infty} c(G)=\infty.$$ That is, I have to show that $\exists $ a function $f:\mathbb{N} \to ...
0
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0answers
11 views

Prove that if there is a surjective homomorphism from $S_n$ onto $C_r$, then $r$ must be $1$ or $2$.

Prove that if there is a surjective homomorphism from $S_n$ onto $C_r$, then $r$ must be $1$ or $2$. I searched this on google, there is a proof using commutator subgroup, unfortunately, i don't know ...
0
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0answers
19 views

A question concerning 4-cycles in $S_4$

Is it true that for all $g\in S_4$ and $f \in S_4$ a 4-cycle, then $g^{-1}fg=h$ implies $h$ is also a 4 cycle. I did a few examples, and it seems to be true, but I don't know how to prove it. Also, I ...
2
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1answer
33 views
1
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1answer
43 views

Prove either $G=ST$ or |$G|\geq|S|+|T|$

Let G be a finite group, and let S and T be (not necessarily distinct) nonempty subsets. prove that either $G=ST$ or |$G|\geq|S|+|T|$ That's my thougt, I am thinking suppose $G$ does not equal to ...
0
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1answer
13 views

Two quotient morphisms and universal property

I am reading some notes on group theory and I am having some doubts related to the following: Let $S \lhd G$ and let $\rho:G \to Q, \space \rho': G \to Q'$ be two quotients of $G$ by $S$. Then, by ...
2
votes
1answer
14 views

Is there a specific name for this set of square-rooted primes?

Consider the set of all the primes numbers (± square rooted) and all the irrational numbers that can be formed under their addition (only the addition of finitely many elements is allowed, i.e. no ...
1
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0answers
24 views

Finding the normal subgroups in a semidirect product

Let the group $A=\cdots\times\mathbb{Z}_{-1}\times\mathbb{Z}_{0}\times\mathbb{Z}_{1}\times\cdots$ with $\mathbb{Z}_{i}=\left\langle a_{i}\right\rangle $ and $\alpha:a_{i}\rightarrow a_{i+1}$ an ...
3
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1answer
54 views

If the commutator of a finite group has order $2$, then the order of the group is divisible by $8$

Prove that if $|G| < \infty$ and $|G'| = 2$ then $|G|$ is divisible by $8$. Thoughts. $A \simeq G / G'$ is abelian and $G' \simeq \mathbb{Z}_2$. Since $G' \subset G$ then at least $|G| \vdots 2$. ...
0
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1answer
13 views

Vector space generated

Let $(L,+)<(\mathbb R^n,+)$ be a additive subgroup and let $\{v_1,\ldots,v_m\}$ be a maximal linearly independent subset of $L$. Let $V$ be the subspace spanned by $\{v_1,\ldots,v_m\}$. Asumme that ...
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1answer
33 views

How to prove there are only finitely many distinct groups with exactly n elements?

How to prove there are only finitely many distinct groups with exactly n elements, If isomorphic groups are regarded as being the same? I set $f: a_{ij}\to b_{ij}$, then it is easily to prove $f$ is ...
2
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0answers
21 views

Generators of group of “unitary” matrices over a finite field

This is about a group related to $U(n,q)$ and $SU(n,q)$. I know from multiple sources the generators for these groups, but $U(n,q)$ is defined to be the group of matrices $A$ such that $A^*JA = J$ ...
1
vote
1answer
21 views

Unique image of torsion groups in the circle group.

Let $p$ be a prime. For any one-to-one homomorphisms $f,g:\Bbb Z_{p^\infty}\to \Bbb T$, we have $f[\Bbb Z_{p^\infty}]=g[\Bbb Z_{p^\infty}]$, where $\Bbb T $ is the circle group. Is this correct ...
2
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2answers
29 views

Clarification on Symmetry Group

My text says "It is a general fact, and an easy one to prove, that the invertible transformations of a mathematical object that preserve some feature of its structure always form a group. We call ...
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votes
3answers
27 views

symmetric difference

I have to prove that the symmetric difference, AΔB = (A∪B) \ (A∩B), is associative for my mathematics study with two inclusions. So I have to prove that: (AΔB)ΔC = AΔ(BΔC) I started with taking an ...
3
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0answers
26 views

homomorphism inducing monomorphism on some quotient group

Let $f:G\rightarrow H$ be a group homomorphism such that $f_* :G_{ab}\rightarrow H_{ab}$ is an isomorphism and that $f_* : H_2(G)\rightarrow H_2(H)$ is an epimorphism. Question is to prove that this ...
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3answers
37 views

Group homomorphism that maps to one element

Is there some way to construct a group homomorphism $G \to H$ that maps everything in $G$ to just one non-identity element of $H$ (besides mapping $0$ to $0$)? For arbitrary (finite) $H$ but you have ...
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0answers
15 views

Transitive Actions, Primitive Actions, and Ergodicity

A group action is transitive iff it has one orbit. Intuitively, this seems to say $G$ shuffles around all the elements of the $G$-set. A group action is primitive iff it has no nontrivial blocks, ...
0
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1answer
32 views

Isomorphisms based on Conjugacy Classes

For what groups of the same order are not isomorphic and contain the same isomorphic class? I as well have a more detailed question: For which of those groups are not abelian. The only example I know ...
14
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3answers
147 views

$|G| + \frac{|G|}{\left|\langle a\rangle\right|} + \frac{|G|}{\left|\langle b\rangle\right|} + \frac{|G|}{\left|\langle ab\rangle\right|}$

Show that for every finite group $G$ and for every elements $a, b \in G$ the following expression $$ |G| + \frac{|G|}{\left|\langle a\rangle\right|} + \frac{|G|}{\left|\langle b\rangle\right|} + ...
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3answers
33 views

What is a conjugacy class of reflection?

I have a problem to do, asking to show that $D_{2n}$ has two conjugacy classes of reflections if n is even, but only one if n is odd. My question is, what is a conjugacy class of reflection? I have ...
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2answers
40 views

Show that $G' = \bigcap_{C\subseteq N \triangleleft G} N$

I am asked to show that $G' = \bigcap_{C\subseteq N \triangleleft G} N$, where $G'$ is the commutator subgroup of $G$, and $C :=\{aba^{-1}b^{-1}\mid a,b\in G\}$. Showing $\bigcap_{C\subseteq N ...
3
votes
2answers
52 views

Groups of Order $n$

Is there a formula for finding the number of groups of order n. For example, if a group $G$ has an order n, is there a formula in which someone can find the number of groups with that order. I suppose ...
2
votes
2answers
31 views

Prove that the group $\mathbb{Z}^{n}$ is generated by at least $n$ elements

I need to prove that the group $\mathbb{Z}^{n}$ with the regular $+$ operation is generated by at least $n$ elements. I know it's pretty analog to the case of vector spaces.. I tried induction ...
-3
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1answer
38 views

A Problem Involving Isomorphism . [on hold]

Please help me with this.. A group theoretic proof that $(\mathbb Q,+)$ is not isomorphic to $(\mathbb R^+,*)$.??
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2answers
31 views

$\mathbb{Q} \simeq \mathbb{Q}^*_+$ isomorphism [duplicate]

Is it true that $(\mathbb{Q},+) \simeq (\mathbb{Q}^*_+, \times)$? If yes then is there any constructive isomorphism?
1
vote
1answer
20 views

Need help understanding this proof that if $H=\{e, (1 \, 2)(3 \, 4), (1\, 3)(2 \, 4), (1 \, 4)(2\, 3)\}$, then it is a normal subgroup of $S_4$.

I'm going to remove some of the details from the proof: If $(i \, j)$ is a transposition in $S_n$ and $\tau \in S_n$, then $\tau (i \, j)\tau^{-1} = (\tau(i) \, \tau(j))$ so for any two disjoint ...
4
votes
1answer
39 views

Fractions with numerator and denominator both odd

Let $$G :=\left\{\frac {a}{b}\in\mathbb{Q}\; ;\; a,b\in\mathbb{Z}, a \text{ odd}, b \text{ odd}\right\}$$ Clearly, $G$ is a subgroup of the multiplicative group $\mathbb{Q}^*$. I was wondering if ...
0
votes
1answer
22 views

“$H_2/G_2$” and “$G_1/H_1$” meaning

I have the following problem: Let $f \colon G_1 \rightarrow G_2$ be an epimorphism, $H_2/G_2$ and $H_1=f^{-1}(H_2)$. Prove that $G_1/H_1 \cong G_2/H_2$. Is this still true if $f$ isn't surjective? ...
3
votes
1answer
63 views

Example of $aH \subsetneq Ha$

Problem. Is there an example of a group $G$, a subgroup $H$ and an element $a \in G$ such that $|G : H| < \infty$ and $aH \subsetneq Ha$?
0
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1answer
41 views

$G$ is a primitive group

Let permutation group $G$ contains a minimal normal subgroup $\neq 1$ which is transitive and Abelian. Show that $G$ is primitive. My attempts: Because of Proposition 4.4. of Wielandt's book ...
0
votes
1answer
41 views

Explanation of Proof that field sum of more than 2 elements is 0. [duplicate]

"Suppose the field $F$ is finite. If $f\colon F\to F$ is any bijection, then we can conclude that $\sum_{x\in F}x=\sum_{x\in F}f(x)$. Let $\alpha\in F$ such that $\alpha\ne 0$. Then $x\mapsto \alpha ...
1
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1answer
39 views

Exact sequence induces exact sequence

Consider exact sequence $N\xrightarrow{f} G\xrightarrow{g} Q\rightarrow 0$ Question is to prove that this gives exact sequence $N/[G,N]\xrightarrow{\bar{f}} G/[G,G]\xrightarrow{\bar{g}} ...
0
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2answers
42 views

Let M and N be normal subgroups of a group G such that $G = MN.$ Prove that $G/(M \cap N) \simeq G/M \times G/N$.

Let M and N be normal subgroups of a group G such that $G = MN.$ Prove that $G/(M \cap N) \simeq G/M \times G/N$. Claim 1: $M\cap N$ is a normal subgroup of G: Proof: $1_G \in M$ and $N$ since M ...
5
votes
2answers
82 views

Product of all elements of a finite group with an unique element of order 2

Well be with you, gentlemen. I have the following problem from Aluffi's Algebra: given a finite group $G$ with an unique element $f$ of order $2$, show that \begin{equation} \prod_{g\in G}g=f ...
0
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1answer
21 views

A problem involving centralizers and order of elements.

Let $a$ be an element of a group $G$ such that $|a| = 5$. Show that $C_G(a)=C_G(a^3)$, where $C_G(a)$ is the centralizer of $a$ in $G$. Also, find an element $a$ of some group $G$ such that $|a|=6$ ...
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2answers
32 views

Infinite Suzuki Groups

Often I found myself on a symbol like $Sz(F)$ where $F$ is an infinite field. What is the definition of an infinite Suzuki group? Are they linear groups? Where I could find some informations about ...
1
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0answers
51 views

what does this image describes about mathematics [on hold]

here I am actually finding the abstract mathematical structures and I got this one also which I did not understand.
3
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1answer
43 views

Find a non-abelian group $G$ and a positive integer m such that for all $g,h \in G$, $(gh)^m = g^mh^m$and $(gh)^{m+1} = g^{m+1}h^{m+1}.$

Find a non-abelian group $G$ and a positive integer m such that for all $g,h \in G$, $$(gh)^m = g^mh^m$$ and $$(gh)^{m+1} = g^{m+1}h^{m+1}.$$ I can't find an example.
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0answers
28 views

How do I prove that H is a normal subgroup?

If $G$ is a group and $a \in G$ but $a \notin H$, where $H$ be a proper subgroup of a group $G$, and if for all $b \in G$, either $b ∈ H$ or $Ha = Hb$, then show that $H$ is a normal subgroup of $G$.
3
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1answer
57 views

Is the following group either a quaternion group or $D_8$?

Let $|G|=2^n$ and $Z(G)=G'=\Phi(G)$ where $\Phi(G)$ is the Frattini subgroup and $|Z(G)|=2$. Is $G$ necassarily either a quaternion group or $D_8$?
1
vote
1answer
39 views

A more swift method for Conjugation Classes

I am asked to find the conjugation classes of a group order n. I am aware what a conjugation class is and how to find it. My question: is there a quicker/more simple way to find the conjugation ...
0
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1answer
31 views

Equality of cosets implies equality of the original sets

Let $H_1$, $H_2$ be two subgroups of $G$ containing $K$, where $K$ is a normal subgroup of $G$. Then if $H_1/K = H_2/K$, prove that $H_1=H_2$. Attempt: Let $h_1K = h_2K$, for some $h_1 \in H_1$, ...
2
votes
1answer
40 views

Rings and Semi-simple rings

I'm failing to see which of the following are semi-simple rings, any help would be appreciated. $\mathbb{C}[X]$, the group ring $\mathbb{Q[Z]}$ and $\begin{pmatrix} \mathbb{Z} & \mathbb{Q}\\ 0 ...
3
votes
2answers
35 views

To prove , if Aut$ (G)$ is trivial then $x^2=e , \forall x \in G$

If for a group $G$ the only automorphism is the identity automorphism , then how do we prove that $x^2=e ,\forall x \in G $ ? I have only been able to prove that $G$ is abelian ; Please Help .
0
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0answers
36 views

Expand group from it's presentation

I want to know if there is a method to expand a group given it's presentation, i.e. list all elements of the group. For instance $G = < x, y \ | \ x^2y = xy^3 = 1>$ (You don't need to solve ...
0
votes
1answer
31 views

What is the center of the Valentiner group $\mathcal{V}=\langle I, Q \rangle$?

(Please refer to this question first: Is $\langle(26543),(34)(56),(12)(3654)\rangle $ isomorphic to $A_6$? ) I want to understand the center of the Valentiner group: $$\mathcal{V}=\langle I, Q ...
2
votes
1answer
18 views

character group of finite abelian group and induced homorphism

This is ex 5.7 of chapter 10 of artin's algebra (2nd edition) Suppose $\varphi:G \rightarrow G'$ is a homomorphism of abelian groups. Define an induced homomorphism $\hat{\varphi}" \hat{G'} ...
0
votes
1answer
35 views

Finding a subgroup of $Z_4⊕Z_2$ that is not of the form $H⊕K$ [on hold]

Find a subgroup of $Z_4⊕Z_2$ that is not of the form $H⊕K$, for $H$ a subgroup of $Z_4$ and $K$ a subgroup of $Z_2$ . Please help
0
votes
1answer
32 views

Additive subgroups of the finite field GF($2^m$)

Consider the set $G=\left\{ {0,1,...,{2^m} - 1} \right\}$. The elements of this set can be viewed as the elements of GF($q=2^m$) with appropriate addition/multiplication operations. For example, GF(4) ...