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

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2
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1answer
57 views

Proving $H \subseteq K$ & $[G:H]=[G:K] $ $\implies$ $H=K$ , without using extended Lagrange's theorem

Let $H \subseteq K$ be subgroups of $G$ such that $[G:H]=[G:K] $ (finite) , then without using the formula $[G:H]=[G:K][K:H]$ , can we prove that $H=K$ ?
0
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0answers
20 views

Irreducible representations of group

I'm basically interested in $C^*$-algebras $A$, where the following conditions for a $^*$-representation $\pi$ on Hilbert space $H$ are all equivalent: 1. $\pi$ is irreducible i.e. there are no ...
2
votes
1answer
70 views

$\operatorname{Aut}(S_4)$ is isomorphic to $S_4$

I already proved this, but I think I can reduce my solution. My solution : There are 4 Sylow 3-subgroup of $S_4$, and denote the set of Syl 3-subgroups by $P=\{P_1,P_2,P_3,P_4\}$. Then, by a group ...
4
votes
1answer
59 views

How can one visualize a homomorphic mapping.

It has been a year or so studying Group theory and Ring theory. Funnily enough, this is the part where i am able to solve most of the questions of the book quite easily, yet not fully understanding ...
2
votes
1answer
43 views

Relation of order of a permutation with its sign

Let $G$ be a group with order $2m$ where $m$ is odd. Consider the left action $\lambda_g:G\to G$. It appears that if $g$ has odd order iff $\lambda_g$ has odd order iff $\lambda_g$ is an even ...
3
votes
1answer
58 views

When is a power of $m$-cycle is also an $m$-cycle?

I have a question taken from Abstract Algebra by Dummit and Foote ($pg.33$, $q.11$): Let $\sigma\in S_{n}$ be an $m$-cycle. Show that $\sigma^{k}$ is also an $m$-cycle iff $\gcd(k,m)=1$. My ...
0
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0answers
54 views

Group theory deep applications [on hold]

I don't have a math degree but I have been learning group theory on my own, starting with getting a conceptual understanding of it using books such as "The Symmetry of Things" by John Conway and ...
1
vote
1answer
39 views

Noetherian group rings

I'm asking for an example of a finitely generated amenable group $G$ and a field $K$, such that the group ring $K[G]$ is not Noetherian. Is it also possible to find a finitely generated amenable ...
2
votes
2answers
131 views

Isomorphism of $(\Bbb{Z}, *)$ and $(\Bbb{Q}-\{0\}, \cdot)$

Are there any operations on $\Bbb{Z}$ that makes it isomorphic to $(\Bbb{Q}-\{0\}, \cdot)$ as a group? Edit: the operation should be made of addition and multiplication of integers, possibly ...
0
votes
1answer
31 views

Show that every imaging f with certain properties is a group

Let $f:\hat{C}\to\hat{C} $ a bijection with the property to sent lines an circles to lines and circles. Show that f is a group with operation the composition of functions (images) (whom obviously ...
3
votes
4answers
43 views

Prove a property about the centralisator

Let G be a group and $U \subseteq G$ a subgroup. Let $x \in G$ be arbitrary. How to show that $C_G(xUx^{-1})=xC_G(U)x^{-1}$ where $C_G(U):=\{g\in G : gu=ug$ $\forall u\in U\}$ For the first ...
2
votes
1answer
44 views

Is this group isomorphic to $S_n\times Z_m^n$?

I have $n$ distinct objects. The objects are ordered in a row, and in each object has an orientation that is a multiple of $2\pi/m$. An action is an arbitrary permutation of the objects, as well as ...
1
vote
1answer
26 views

The no. of homomorphisms between third roots of unity and $S_3$

Find the number of homomorphisms between $(\{1, \omega, \omega^2\}, \times)$ and $S_3$. ($\omega$ being one of the third roots of unity). Apart from the trivial homomorphism we have another ...
2
votes
2answers
48 views

Restriction of a group homomorphism to a normal subgroup

Suppose $f:G\to G$ is a group homomorphism and let $N\trianglelefteq G$. What can we say about $f$ if restriction of $f$ to $N$ is an identity on $N$? Can we say anything "nice" in this situation.
0
votes
0answers
65 views

Canonical topology on standard groups?

I just wanted to know whether there is any standard topology on groups like $\mathbb{Z}/n\mathbb{Z}$ or $\mathbb{Z}$ ? - The only one that I could imagine, especially for finite groups is the discrete ...
128
votes
1answer
3k views

Is there a 0-1 law for the theory of groups?

For each first order sentence $\phi$ in the language of groups, define : $$p_N(\phi)=\frac{\text{number of nonisomorphic groups $G$ of order} \le N\text{ such that } \phi \text{ is valid in } ...
1
vote
2answers
36 views

$a,b$ are elements of the group $G$. Why $|ba|\leq |ab|$ in the following scenario?

A scenario: The order, $n$, of $ab$ (and hence, by definition, the order of the cyclic subgroup $\langle ab\rangle$) is finite (thus, the order of $ba$ is finite). Then $(ab)^n=e$. So ...
0
votes
1answer
35 views

Repetition of elements in a group, where am I wrong?

I am employing a particular method by which I am getting absurd results. I can not figure out where I am mistaking. Say $G$ is an abelian group of order 9. Let $a \in G$, so I am representing the ...
2
votes
1answer
208 views

Möbius transformations forming a group and isomorphism with $S_{3}(D_{6})$

My task is to prove that the Möbius transformations defined by $z,\frac{1}{z},1-z,\frac{1}{1-z},\frac{z}{z-1},\frac{z-1}{z}$ make up a group that is isomorphic to the group $S_{3}(D_{6})$. Identify ...
0
votes
2answers
263 views

Nonabelian group of order $p^3$ and semidirect products

Let G be a nonabelian group of order $p^3$ where p is an odd prime. Suppose that G contains an element of order $p^2$. Then G is isomorphic to the semidirect product $Z_{p^2} \rtimes_{\alpha} Z_p$, ...
2
votes
2answers
78 views

The difference between vector space and group

When comparing the difference between the definition of vector space, I see that the main job is that vector space defines a scalar product while the group not, so here list two of my questions? ...
2
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0answers
43 views

Isomorphisms between semi-direct products

Let $H$ be any group and $K$ an abelian group. (I'm interested in $K={\mathbb Z}$.) Homomorphisms $H\to Aut(K)$ define semi-direct products $K\rtimes H$. There is an action of $Aut(H)\times Aut(K)$ ...
2
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2answers
38 views

If G is a group and N is normal in G with index d, then $x^d \in N$

I want to show the statement in the title. If $G$ is a group and $N$ is normal in $G$ with $[G:N]=d$, then $x^d \in N$ for all $x \in G$ I want to consider the image $xN$ of $x$ in $G/N$ $G/N$ has ...
0
votes
1answer
33 views

pemutation representation that confuses me a lot recently

For any group G, define group action on a set A. There will be a permutation representation of that group action. I am kind of confused why the permutation representation can be used to reflect the ...
3
votes
0answers
28 views

Show that Icosahedral group does not have normal subgroup of order 5

Here is my work so far: Let $G$ denote the group of orientation-preserving isometries of Icosahedron. I have shown that $G$ acts transitively on $G/G_s$ where $G_s$ is the isotropy group, namely ...
1
vote
2answers
29 views

Do two sets generate the same subgroup/normal subgroup?

Suppose I have a group $G$, given as finitely many generators and finitely many relations, say. Furthermore, suppose I have two finite sets $A,B$ of elements of the group. I would like to know if the ...
2
votes
2answers
62 views

How to find $[A_n,A_n]$

Let $n \in \mathbb{N}$. How could I find $$ [A_n,A_n] \quad \cong \quad \langle ghg^{-1}h^{-1} \ : \ g,h \in A_n \rangle $$ My own thoughts I remembered that any element in $A_n$ can be written as ...
1
vote
1answer
56 views

Whether two quotients of $\mathbb{Z}^2$ are isomorphic.

Let $H_1$ be the subgroup of $\mathbb{Z}^2$ generated by $\{(1,2),(4,1)\}$, let $H_2$ be the subgroup of $\mathbb{Z}^2$ generated by $\{(3,2),(1,3)\}$. Is it true that $\mathbb{Z}^2/H_1\cong ...
3
votes
1answer
52 views

If $p$ is prime, prove that $\exists k\in\lbrace 5,-7,9,-11,..\rbrace$ in $(\mathbb{Z}/p\mathbb{Z})^*$ so that the Legendre symbol $(\frac{k}{p})=-1$

The BSPW primality test, when given $p$ as input, iterates over $k \in \lbrace 5,-7,9,-11,...\rbrace$ as long as the Legendre symbol $(\frac{k}{p})=1$. If $(\frac{k}{p})=0$, it returns "composite". So ...
3
votes
1answer
43 views

The order of permutation groups and alternating groups

The question was: True or False: $\forall{n}\in{\mathbb{N}}$ the group $S_n$ and $A_n$ have different sizes. My answer is False. That is since both $A_1 =(\text{id})$ and $S_1 =(\text{id})$. ...
5
votes
4answers
132 views

Finitely-generated group such that all (non-trivial) normal subgroups have finite index implies all (non-trivial) subgroups have finite index?

Let $G$ be a finitely generated group such that every non-trivial normal subgroup has finite index. Does it follow that every non-trivial subgroup of $G$ has finite index? This question arose as ...
0
votes
1answer
45 views

Group Theory- S3 table

$\begin{matrix} & e& a& b& c& d&f \\ e& e& a & b& c& d&f \\ a& a& b& e& d& f&c \\ b& b& e& ...
0
votes
0answers
80 views

Sylow subgroups

Let $G$ be a groups of order $385$. 1. Show that $P_7$,$P_{11}\triangleleft G$. 2. Show that $P_7 \subseteq Z(G)$. 3. Show that $Z(G)=P_7$ or that $G$ is cyclic. *for $P_i$ - the Sylow-$i$ ...
0
votes
1answer
21 views

On a maximal subgroup of the direct product of groups

Let $A$ be a maximal abelian (nilpotent) subgroup of the direct product $G=F\times H$ of groups $F$ and $H$.Then prove that $$A= (A\cap F)\times (A\cap H).$$
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vote
3answers
49 views

A subgroup such that at least one left coset is a right coset

I saw an exercise that goes: Let $G$ be a group of order 120, and $H$ be a subgroup of order 24. If at least one left coset of $H$ in $G$ is a right coset apart from $H$ itself, show that $H$ is ...
4
votes
1answer
63 views

Another Presentation of Certain Cyclic Groups

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^n\rangle $$ is cyclic of order $3(n+1)$, for $n=0 \mod 3$ or $n= 1 \mod 3$, $n\ge 0$. This ...
5
votes
6answers
213 views

A presentation of a group of order 12

Show that the presentation $G=\langle a,b,c\mid a^2 = b^2 = c^3 = 1, ab = ba, cac^{-1} = b, cbc^{-1} =ab\rangle$ defines a group of order $12$. I tried to let $d=ab\Rightarrow G=\langle d,c\mid ...
-1
votes
3answers
48 views

$G$ is an abelian group of order a product of distinct primes $\implies G$ is cyclic?

If $G$ is an abelian group of order $p_1p_2...p_k$ , where $p_1,p_2,...,p_k$ are distinct primes , then is it true that $G$ is cyclic ?
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2answers
74 views

Must the centralizer of a non-identity element of a group be abelian?

Q1: Must the centralizer of a non-identity element of a group be abelian? I challenge everyone by asking further about this question one step further. The definition of centralizer is: Let a be ...
6
votes
3answers
96 views

Automorphisms of $\mathbb{Z}/p\oplus\cdots\oplus \mathbb{Z}/p$

Consider the abelian group $$G = \underbrace{\mathbb{Z}/p\oplus\cdots\oplus \mathbb{Z}/p}_{n},$$ where $p$ is prime and $1\le n \le p$. I want to show that $G$ has no automorphism of order $p^2$. I ...
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votes
0answers
24 views

quasiprimitive non-solvable groups

I'm looking for the reference about quasiprimitive unsoluble groups. Actually we can find a lot of useful things about quasiprimitive solvable groups in "Representations of solvable groups by Manz and ...
0
votes
3answers
560 views

What does it mean for a group to be Abelian?

I'm revising questions on groups for exams, and I still can't quite understand what an Abelian group is. Please help me understand, if anyone could give me a more simple explanation.
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2answers
59 views

Normal subgroup created by a bunch of elements

if I have a finite group $G$ and a bunch of elements that are the elements of a set $A$. How can I systematically calculate the smallest normal subgroup of $G$ that contains $A$? I am rather ...
4
votes
2answers
92 views

In group theory, is it true that $f(X \vee Y) = f(X) \vee f(Y)$?

($\vee$ denotes join). Let $G$ and $H$ denote Abelian groups, $X$ and $Y$ denote subalgebras of $G$, and let $f : G \rightarrow H$ denote a homomorphism. Then: $$f(X \vee Y) = f(X+Y) = f(X)+f(Y) = ...
1
vote
1answer
26 views

Schur Multiplier of general linear group

Ideally I would like to know the Schur multiplier of $Gl(n, F_3)$, but perhaps this is not reasonable to ask. But for a small fixed $n$, this should be known, but i could not find any result when ...
1
vote
1answer
38 views

Is my proof correct? (about commutators)

I want to prove the following fact: Let $G$ be a finite group and let $x, y\in G$ such that $[x, y] \in Z(G)$. Then $[x, y^s] = [x, y]^s$ for every $s\in \mathbb{Z}$. If we assume that $[x^r, y^s] ...
4
votes
2answers
203 views

Cyclic Group Presentation

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^7\rangle $$ is cyclic of order 24. This presentation was obtained using the Todd-Coxeter ...
0
votes
0answers
23 views

Homeomorphism between SU(4) and SO(6)

http://www.mat.univie.ac.at/~westra/so3su2.pdf said that $\mathrm{SU}(2)$ acts homeomorphism to $\mathrm{SO}(3)$, via $$ \begin{pmatrix} z & w \\ -\bar w & \bar z \end{pmatrix} \mapsto ...
3
votes
4answers
117 views

Examples of Infinite Simple Groups

I would like a list of infinite simple groups. I am only aware of $A_\infty$. Any example is welcome, but I'm particularly interested in examples of infinite fields and values of $n$ such that ...
2
votes
2answers
45 views

Determine the center of this finitely presented group.

Consider the group $G(n)= \langle a, b \ \vert\ aba^{-1}=b^{n+1}, bab^{-1}=a^{n+1} \rangle$, $n\ge 1.$ Show that the center $Z$ is cyclic of order $n$ and that $G/Z$ is abelian of order $n^2$. This ...