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

learn more… | top users | synonyms (2)

2
votes
1answer
11 views

What could be said about $U,V$ if $UN = VN$ for some $N \unlhd G$.

Let $N \unlhd G$ and let $U, V$ be two subgroups, if $UN = VN$, is it possible that $U \ne V$ if i) $U$ and $V$ are not contained in $N$, and ii) if $U\cap N = V\cap N = 1$. Of course, if $U, V \le ...
2
votes
1answer
31 views

Must a $R$-automorphism on $R[X]$ be of the form $X\mapsto aX+b,\ a\in R^*,b\in R$?

Let $R$ be a commutative ring, I wonder if every $R$-automorphism (that is, a ring automorphism that fix $R$) $\varphi$ on $R[X]$ satisfies $\varphi(X)=aX+b$, where $a$ is an unit in $R$ and b an ...
0
votes
1answer
54 views

Does every group have a generator?

Can't ask my question specifically because it's on a take-home test, but it's critical to my current proof that I'm able to pick a generator g from a group G, and I'm not sure if I can do that.
0
votes
2answers
22 views

Orders of elements in alternating group $A_8$

I have an issue with a question from some homework for my introduction to group theory course. For which integers d does the alternating group $A_8$ have elements of order d? So through some ...
2
votes
1answer
69 views

To prove set is a group

Given a non empty set together with associative binary operation $*$ on $G$ such that $a*x=b$ and $y*a=b$ have solutions in $G$ for all $a,b$ in $G$ To prove it is a group Hints to get started ...
-1
votes
1answer
27 views

properties of alternating subgroup?

I was wondering, is it true that if $Alt_n$ is an alternating subgroup of $Sym_n$ for $n>3$, $Alt_{n-i}\leq Alt_n$ for all $i<n$?
0
votes
0answers
81 views

Why is the number of conjugacy classes modulo 16 equal to the order for a finite group of odd order? [on hold]

Let $G$ be a group of odd order $n$ and suppose $|Con(G)| = k$, prove that $$k \equiv n \pmod{ 16}.$$ How do I proceed on this? Thanks.
-1
votes
1answer
19 views

Centre of matrixgroup $GL_2(\mathbb{R})$ [duplicate]

What is the centre of the matrixgroup $GL_2(\mathbb{R})$? Edit: I didn't know this was a duplicate
-1
votes
0answers
27 views

Indecomposable Finite Abelian Groups are just Cyclic Groups of a Prime Power Order.

there is a structure theorem in group theory states that: indecomposable finite abelian groups are just cyclic groups of a prime power order. my question is about proving this theorem. how to prove? ...
-3
votes
0answers
33 views

Centre $Z(S_n)$ of $S_n$ trivial for $n\neq 2$ [on hold]

How do I prove that the centre $Z(S_n)$ of $S_n$ is trivial for $n\neq 2$? Edit: What is $Z(S_2)$?
0
votes
0answers
157 views

What are the properties of the topological group $e^{i\mathbb{Q}}$?

What are its properties as a topological group? It is not $\mathbb{Q}/\mathbb{Z}$ but resembles it, it contains the subgroups $e^{i\mathbb{Z}}\supseteq e^{ip\mathbb{Z}}\supseteq ...
1
vote
1answer
57 views

Are isomorphisms always constructable?

Suppose we are given that two finitely presented groups are isomorphic. Is it possible to construct an isomorphism between them? More precisely, you are given $G_1=\langle S\rangle\cong G_2$, $S$ a ...
0
votes
0answers
26 views

Properties of p-residue group

Related thread (definition of $O^{p'}(G)$) : does minimality condition imply normal p-sylow subgroup > Assume that $G$ is a finite group, and that $p$ is a prime number dividing the order of $G$. ...
0
votes
2answers
36 views

What's the notation for the intersection of stabilizer subgroups on a subset?

Let $G$ acting on the the (finite) set $S$, or the (finite dimensional) space $V$. Let $s \in S$, then the stabilizer $G_s:= \{ g \in G \ \vert \ gs = s \}$. Let $R \subset S$, then there are ...
1
vote
1answer
53 views

Show that H is normal subgroup of G. [duplicate]

Let H be a proper subgroup of G and a $\in$ G, a $\notin$ H. Suppose that for all b $\in$ G, either b $\in$ H, or Ha = Hb. Show that H is normal subgroup of G. How do I proceed on this?
0
votes
0answers
29 views

comparing the size of cosets

A,B are subgroups of G, and $|G:A|< \infty$. so I am trying to show two things: a) show that $|A:A\cap B|<|G:A|$. b)$|A:A \cap B| = |G:A|$ iff AB=G. For a, my attempt is to construct map from ...
0
votes
0answers
19 views

Why is $E[l]\cong\mathbb Z/l\mathbb Z\times\mathbb Z/l\mathbb Z$ for an elliptic curve $E$?

René Schoof's 1995 paper contains the following statement about an elliptic curve $E$ (at the bottom of page 233): [...], we use the subgroup $E[l]$ of $l$-torsion points of $E(\overline{\mathbb ...
1
vote
2answers
34 views

Why is the permutation $(a,c,d,e)(a,b)=(a,d)(b,c,e)$

I'm working through a proof in my notes. We already know that the transposition $(a,b)\in G$ and $(a,b,c,d,e)\in G$, where $G$ is a group of permutations of the elements $a,b,c,d,e$, so it's a ...
0
votes
0answers
18 views

Automorphisms of B_n

Consider the Coxeter group of type $B_n$. This group, of order $2^n n!$, can be identified with the group of odd permutations of the set $\{\pm 1,\dots,\pm n\}$ and is thus isomorphic to the ...
0
votes
4answers
49 views

$(G, \cdot)$ - group such that $|G|<\infty$. $G$ has exactly $2$ subgroups

$(G, \cdot)$ - group such that $|G|<\infty$. $G$ has exactly $2$ subgroups. Prove that order of $a\in G$ is a prime if $a\neq e$ where $e$ is neutral element. I am stuck. Here is what I got so ...
0
votes
0answers
36 views

Basic questions regarding group theory and symmetry in practice

I have for some months been interested in group theory. I was very fascinated by the level of abstraction I first met when working with groups. Another aspect that has fascinated me lately is ...
-1
votes
0answers
20 views

Counting homomorphisms

Give the number of homomorphisms from $D_5$ to $R*$ (R under multiplication). Do the same for $D_5$ to $S_4$ Two of my homework problems, no idea how to start.
0
votes
0answers
17 views

Divisibility of orders on a group

Let $h \in H \leq G$, where $G$ is a finite group and $H$ is a subgroup. From Lagrange's Theorem, we know $o(h)$ divides $|G|$. It is still true for $H$? That is, is $o(h)$ going to divide $|H|$? I ...
0
votes
1answer
21 views

Proof about the Sylow $2$-subgroups of permutation group such that each element has at most two fixed points

Let $G$ be a finite, transitive, nonregular permutation group on $\Omega$ such that every element of $g \in G^{\#} := G \setminus \{ 1_G \}$ has at most two fixed points. Suppose further that ...
-4
votes
0answers
28 views

Ring and group theory [on hold]

Let $R$ be a ring and let $J$ be a left ideal of $R$. a) Let $K$ be a left ideal of $R$. Show that $(JK)^n$ subset and equal of $J^n K$ for every $n \in \mathbb{N}$ ( it can be done with or ...
1
vote
2answers
38 views

Let $G$ be a finite group and $H < G$. Prove that $n_p(H) \le n_p(G)$.

Let $G$ be a finite group and $H < G$. Prove that $n_p(H) \le n_p(G)$. Ok, so now i know that $n_p(H) \le n_p(G)$ refers to the number of Sylow p-subgroups in H and G, respectively. From here, I ...
3
votes
0answers
25 views

About the construction of resolvents in Galois theory (over $\mathbb{Q}$ in $\mathbb{C}$)

I have to say that my question is quite long and I apologize for this. The main idea is that I would like to show how to construct resolvents for any transitive subgroup of the permutation group to ...
0
votes
2answers
27 views

Showing that the group is abelian

Let $\sigma = (123456)$ in $S_6$. And let $G = \{e, \sigma, \sigma^2, \sigma^3, \sigma^4, \sigma^5\}$ be a group under operation from $S_6$. Is $G$ abelian? Workings: A group is abelian if it is ...
0
votes
0answers
30 views

How to check if a number is in $\mathbb{Z}_{p}^{\star}$, where $p$ is the product of two primes?

Like the question says, how can I check if a given number $e$ is in the group $\mathbb{Z}_{p}^{\star}$, where $p$ is the product of two primes? It is enough to verify if $\gcd(e, p)=1$? What's the ...
0
votes
1answer
37 views

A question about free product $\mathbb{Z_{2}}*\mathbb{Z_{2}}*\mathbb{Z_{2}}$

I want to use some examples to comprehension the definition of free product. Let $\mathbb{Z_{2}}$ be the integers $\{o, ...,m-1\}$ with addition modulo $m$ as the group operation and ...
2
votes
2answers
51 views

Outer automorphism of $S_6$ and conjugate stabilizers

Let $f:S_6 \mapsto S_6$ be an outer automorphism of $S_6$ and consider the subgroups $$G = \{\pi \in S_6 \mid \pi(1) = 1\}$$ and $$H = \{\pi \in S_6 \mid f(\pi)(1) = 1\}.$$ I would like to show that ...
-1
votes
0answers
39 views

To write right cosets of H in G [on hold]

Let $G = A(S)$, the set of all bijections of $S$, where $S= \{x_1,x_2,x_3\}$, and let $H = \{\sigma \in G\ | (x_1)\sigma = x_1\}$ (here, $(x_1)\sigma$ denotes the image of $x_1$ under $\sigma$). ...
1
vote
1answer
18 views

To find right cosets of H in G where G=<a> and H=$<a^{2}>$ ,where o(G)=10

To find right cosets of H in G where G= and H=$<a^{2}>$ ,where o(G)=10 Since order of $G =10$ , so $a^{10}=e$ .We have $G= { a,a^{2},a^{3},a^{4},a^{5},a^{6},a^{7},a^{8},a^{9},e}$ and $H = ...
1
vote
1answer
42 views

If G has no non trivial subgroups ,then Show that G must be of prime order

If G has no non trivial subgroups ,then Show that G must be of prime order .This question is from Herstein Page 46 Question 3 . Attempt :- Let G has prime order(say p) .So by Lagrange theorem ...
3
votes
1answer
30 views

Kernel of a homomorphism map

Let $\pi : \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by $(x,y)\mapsto x + y$. Describe the kernel and fibers of $\pi$ geometrically. My attempt: Let $x,y,z$ and $w$ be in $\mathbb{R}$. Since ...
1
vote
1answer
36 views

do any two conjugates of [[1,1],[0,1]] generate SL(2,Z)?

Do any two (distinct) conjugates of the matrix $[[1,1],[0,1]]$ generate $SL(2,\mathbb{Z})$? Of course the conjugates $[[1,1],[0,1]]$ and $[[1,0],[1,1]]$ generate it, and some explicit computations ...
-1
votes
0answers
27 views

Is there any way to figure out how many orbits there are? [on hold]

Let G=A4 act on itself by conjugation I want to find out orbits of this group Is it possible to find out how many orbits there are without actual computation?
1
vote
2answers
72 views

$Aut(D_4)$ is isomorphic to $D_4$?

Problem statement: I need to find out if $Aut(D_4)$ is isomorphic to itself($D_4$) and explain my answer. I already know that it is isomorphic, so now all I need to do is to prove it. I assume that ...
4
votes
1answer
28 views

Nilpotent and solvable groups

If $G$ is a finitley generated group say by $x_1,\ldots,x_n$, then $G$ is abelian if $x_i$ and $x_j$ commute for every $i,j$. This measn we can check if $G$ is abelian by just looking at the ...
-1
votes
1answer
22 views

If $G$ and $N$ are soluble, then $G/N$ is soluble

Let $N$ be a normal subgroup of $G$. If both $G$ and $N$ are soluble, then $G/N$ is also soluble. My attempt: $G$ is soluble, so there exists a subnormal series $1=G_0\subset ... \subset G_n=G$. I ...
0
votes
3answers
25 views

If $\phi :G\rightarrow H$ is a group homomorphism and $G$ is soluble, then $Im(\phi)$ is also soluble

I'm trying to prove the following statement: If $\phi :G\rightarrow H$ is a group homomorphism and $G$ is soluble, then $Im(\phi)$ is also soluble, I tried creating a map $\psi:G\rightarrow ...
0
votes
4answers
188 views

To prove every element of G has finite order where

Let G be a group such that intersection of all its subgroups which are different from e is a subgroup different from e . To prove every element of G has finite order Hints to get started Thanks ...
1
vote
1answer
30 views

Is $A_n$ isomorphic to $C_n$ in general

Let $A_n$ denote the alternating group of order $n$, and let $C_n$ be the cyclic group of order $n$. Correct me if I'm wrong, but I know that $A_3 = \{(),(1,2,3),(1,3,2)\}\cong \{0,1,2\}=C_3$. I'm not ...
0
votes
2answers
29 views

To prove $o(HK) = o(H)o(K)/o(H\cap K)$

Given that $H$ and $K$ are finite subgroups of $G$ of order $o(H)$ and $o(K)$, prove that $$o(HK) = \frac{o(H)\,o(K)}{o(H\cap K)}$$ I have proved for specific case when $H$ and $K$ have only ...
1
vote
1answer
51 views

Products and relationships of ideals of Ring R.

Let $R$ be a ring and let $I$ be a left ideal of $R$. (a) Let $K$ be a left ideal of $R$. Show that $(IK)^{n} \subseteq I^{n}K$ for all $n \in \mathbb{N}$ (b) Show that $I+ IR$ is a two-sided ideal ...
1
vote
2answers
64 views

Enumeration of Solved Sudoku puzzles

I tried asking this on StackOverflow and it was quickly closed for being too broad, so I come here to get the mathematical part nailed down, and then I can do the rest with no help, most likely. From ...
3
votes
1answer
35 views

Two Lemmata about permutation groups such that every element has at most two fixed points

Let $G$ be a finite, transitive, nonregular permutation group on $\Omega$ such that every element of $g \in G^{\#} := G \setminus \{ 1_G \}$ has at most two fixed points. Suppose further that ...
1
vote
1answer
57 views

Difference between H/N, HN and H∩N

I'm trying to understand the second isomorphism theorem, and I'm kind of stuck. An example in Z would be very appreciated. :)
7
votes
1answer
50 views

Applications of the following theorem in the real world

We know that every permutation can be expressed as a product of transpositions ( cycles with length 2). As a class project I'm looking for the applications of this fact in the real world; especially ...
1
vote
0answers
31 views

Group Action determined by equations

Let $G$ a group, $X$ a set. An action of $G$ on $X$ is given by a mapping $(g,x) \to g \cdot x$, which satisfies $g \cdot (h \cdot x) = (gh) \cdot x$ and $e_G \cdot x = x$. In other words, to ...