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

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14
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3answers
174 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|} + ...
0
votes
3answers
37 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 ...
1
vote
2answers
41 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
58 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
33 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
votes
1answer
38 views

A Problem Involving Isomorphism . [closed]

Please help me with this.. A group theoretic proof that $(\mathbb Q,+)$ is not isomorphic to $(\mathbb R^+,*)$.??
0
votes
2answers
32 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
21 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
41 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
24 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
65 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
votes
1answer
45 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
43 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
vote
1answer
46 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
votes
2answers
49 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
85 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
votes
1answer
22 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 $|a|=6$ ...
1
vote
2answers
33 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
vote
0answers
67 views

how is this image related to group theory? [closed]

here I am actually finding the abstract mathematical structures and I got this one also which I did not understand.
3
votes
1answer
47 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.
0
votes
0answers
30 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
votes
1answer
62 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
40 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
votes
1answer
34 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
43 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
37 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
votes
0answers
38 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 ...
1
vote
1answer
34 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
34 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
36 views

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

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
34 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) ...
4
votes
1answer
52 views

every group of order $105$ has a cyclic normal subgroup of index $3$ ?

Does every group of order $105$ has a cyclic normal subgroup of index $3$ ? (Please don't use Sylow theorems )
2
votes
1answer
55 views

A question in Abstract Algebra about cosets

I tried to solve this problem but without success: Let $H$ be a subgroup of a group $G$. Build an injective function from $G/H$ (the set of left cosets of $H$) to $H\setminus G$ (the set of right ...
0
votes
2answers
35 views

Two homomorphisms must be equal if $ \phi (a) = \psi (a) $

Given $G_1$, a cyclic group, and $G_2$ a group with $\phi$ and $\psi$ homomorphisms from $G_1$ to $G_2$ how can I show that $\phi=\psi$ if and only if $\phi(a)=\psi(a)$
0
votes
0answers
31 views

group of order $36$ abelian

Let $G $ be a group of order $36$.How to conclude whether it is abelian or not .I tried using Sylow's theorems by calculating the number of subgroups of order $4$ and $9$ but I am getting so many ...
0
votes
2answers
26 views

Let H be a proper subgroup of G of order prime $p^k$ and $N(H) = \{a \in G|aHa^{-1} = H\}.$Show that $N(H) \neq H.$

Let G be a group of order $p^k$ where p is a prime and k is a positive integer. Let H be a proper subgroup of G and $$N(H) = \{a \in G|aHa^{-1} = H\}.$$ Show that $$N(H) \neq H.$$ I think I need to ...
1
vote
1answer
15 views

Internal and Direct Product question, need help with explanation

I am asked to express a group G={1,7,17,23,49,55,65,71} under multiplication modulo 96 as an external and internal direct product of cyclic groups. However I also have an example to help with it, but ...
3
votes
1answer
71 views

Is $\langle(26543),(34)(56),(12)(3654)\rangle $ isomorphic to $A_6$?

My question is the one in the title. If you want to understand the context of the problem, please read further. I reduced a problem to proving the question. Background is: Valentiner group ...
0
votes
0answers
38 views

Isomorphism from $Aut(\mathbb{Z}_2 \times \mathbb{Z}_4)$ to $D_8$

I want to show that $Aut(\mathbb{Z}_2 \times \mathbb{Z}_4) \simeq D_8$ (isometry group of square) I' have got an idea to show isomorphisms : $f: Aut(\mathbb{Z}_2 \times \mathbb{Z}_4) \rightarrow ...
0
votes
2answers
27 views

Infinite order group that has no nontrivial subgroup?

Is there any infinite order group that has no nontrivial subgroup? I guess there isn't, but I don't know how to approach.
1
vote
1answer
33 views

Transitive Actions

Given that transitive actions are in a bijection with conjugacy classes of subgroups of G, describe isomorphism classes of transitive actions for the following groups: $C_4, Z/8, C_2 × C_2, S_3$ Can ...
1
vote
1answer
31 views

Proof of the existence of inverse elements for a group

The group to be determined is defined as follows: $\{x\in\Bbb{Z^4}:x_1x_4=1+x_2x_3\}$ with $(x,y)\mapsto(x_1y_1+x_2y_3,x_1y_2+x_2y_4,x_3y_1+x_4y_3,x_3y_2+x_4y_4)$ $*$ denotes the operation. We have ...
0
votes
2answers
29 views

2 problems with group actions on a finite set.

I've got two problems, which I can't solve on my own so I ask you to help me a little, with some tips :) Prove, that every action of group ( order 9 ) on a set of 8 elements has a fixed point. Show ...
1
vote
2answers
21 views

Examples of group homomorphisms with isomorphic but not equal images

This may be a poor question. I am having trouble thinking of a pair of group homomorphisms: $\varphi, \Psi: G \rightarrow H$ between groups where $\varphi(G) \neq \Psi(G)$ but $\varphi(G) \cong ...
1
vote
2answers
35 views

Homomorphism preserves normality

let $\phi:G\rightarrow G'$ be a homomorphism, and let N' be a normal subgroup of G'. I want to show that $\phi^{-1}[N']$ is also normal subgroup of G. My work : since homomorphism preserves ...
0
votes
2answers
28 views

Determine, whether the following sets together with the mappings are groups or not

I'm a newbie in groups and currently working on some exercises to get familiar with the material. I have to determine, whether the following sets together with the mappings are groups or not. I have ...
-1
votes
1answer
36 views

Alternate Form of Correspondence Principle

Let G be a group and K a normal subgroup in G. Let f : G → G/K be the canonical epimorphism given by x 􏰁→ xK and L a subgroup of G/K. Then (1) There exists a subgroup H of G. H contains K, and ...
0
votes
1answer
51 views

Is infinite product of Z a group?

Having the usual coordinate-wise addition, does infinite product of $\mathbb{Z}$ forms a group? $a,b\in \prod ^\infty \mathbb{Z}$ $a=(a_1,a_2,...)$ $b=(b_1,b_2,...)$ $a\circ ...
-2
votes
0answers
31 views

whether there are some solutions about the exercises in Rotman's An Introduction to The Group Theory? [closed]

whether there are some solutions about the exercises in Rotman's An Introduction to The Group Theory? I feel sick when I can't check whether my solution is correct.T.T
2
votes
2answers
49 views

Groups of order 24.

I supposed $n_3=4$ and $n_2=3$, and then I made $G$ act by conjugation on $Syl_3 (G)$. I want to show that $G\cong S_4$ (looking at all order 24 groups here ...