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

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3
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1answer
94 views

Any subgroup of an abelian group is undistorted.

I need some help with the following math problem. I am studying some notes on Geometric Group Theory and I came across the following problem. Prove: Any subgroup of an abelian group is ...
2
votes
1answer
29 views

Extension theory and automorphism extension

My question is motivated by the following two posts On finite 2-groups that whose center is not cyclic and Automorphisms of group extensions Question: Assume that $A,B,C$ are there algebraic ...
2
votes
2answers
58 views

Cardinality of Centralizer of some element

Let $G$ be a finite group. How can we show that $|G/G^{'}|\leq |C_G(x)|$ for all elements $x\in G$?
2
votes
1answer
48 views

Groups with order $p^3$ ($p$ prime) have two non commutative isomorphism classes

I read in an exercise, that a group with $p^3$ ($p$ prime) elements have $2$ non commutative isomorphism classes. Unfortunately there was just this statement without any explanation. We just solved it ...
1
vote
1answer
34 views

Show that $|5|=2^{n-2}$ when $n \ge 3$ in $U(2^n)$

Show that $|5|=2^{n-2}$ when $n \ge 3$ in $U(2^n)$. I wrote ...
0
votes
0answers
20 views

How to find out exactly invariant factor decomposition of finitely generated abelian groups

Suppose that we defined some finitely generated abelian group $G$. Now how does one find invariant or primary decomposition of $G$? We know that decomposition exists, how do we exactly state ...
4
votes
1answer
65 views

On finite 2-groups that whose center is not cyclic

Let $G$ be a finite 2-group such that $\left|\dfrac{G}{Z(G)}\right|=4$, $Z(G)$ is not cyclic and $Z(G)$ has at least one element of order 4. Then prove that there exists an automorphism $\alpha$ of ...
4
votes
2answers
50 views

A question on terminology (Group Theory)

Is there a standard adjective to describe a finite group $G$ of composite order which possesses, for each (positive) divisor $d$ of $|G|$, a subgroup of order $d$? I would guess "Lagrangian" but I ...
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4answers
49 views

A finite group of order $n$, having a subgroup of order $k$ for each divisor $k$ of $n$, is not simple?

I was asked to prove that, if a finite group $G$ of order $n$ has a subgroup of order $k$ for each divisor $k$ of $n$, then $G$ is not simple. I tried to do this but I could not. Can anyone please ...
7
votes
1answer
115 views

Showing that group of orientation preserving isometries of Icosahedron is a simple group

Let $G$ denote the group of orientation preserving isometries of Icosahedron. To prove the claim, I have shown that $\nexists \ N \ \triangleleft \ G$ such that $|N|=5.$ $\nexists \ N \ ...
6
votes
3answers
588 views

Show group of order $4n + 2$ has a subgroup of index 2.

Let $n$ be a positive integer. Show that any group of order $4n + 2$ has a subgroup of index 2. (Hint: Use left regular representation and Cauchy's Theorem to get an odd permutation.) I can easily ...
0
votes
0answers
30 views

If the automorphism group of a group is cyclic, then the group is commutative [duplicate]

Let $G$ be a group and the $Aut(G)$ group is cyclic $\Rightarrow$ the group $G$ is commutative. I looked at the homomorphism $\varphi : G \rightarrow Aut(G) \ g \mapsto (x \mapsto gxg^{-1})$. Let ...
1
vote
0answers
36 views

Let Cay(G, S) be the cayley graph of G with respect to the finite generating set S where G=⟨S∣R⟩ and R is finite.

Let $\operatorname{Cay}(G, S)$ be the cayley graph of $G$ with respect to the finite generating set $S$ where $G = \langle S\mid R\rangle$ and $R$ is finite. I am reading some notes that claim that ...
1
vote
1answer
84 views

Geometrically describe these Cosets and form a bijection with the Orbit-Stabilizer relation.

I am beginning to study abstract algebra/group theory and I have some seemingly simple practice questions here. I just want make sure I am understanding the concepts correctly. Here are the questions: ...
2
votes
0answers
37 views

On group-theoretic shorthand notation

I have often seen shorthand notation used in group-theoretic contexts and I believe it is called ATLAS notation. However, even with some searching I have not been able to find a satisfactory summary ...
1
vote
1answer
68 views

The unofficial definition of Group.

As in the Visual Group Theory Book the unofficial definition of a group says that : A group is a collection of actions satisfying the rules: 1. there is a predefined list of actions that never ...
2
votes
2answers
78 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
2answers
82 views

Show that $Z\times Z$ is not cyclic… [duplicate]

The full problem is as stated in the title. I am here to check if this is a valid proof. I thought it would be easiest using Linear Algebra. Recall that an infinite cyclic group is isomorphic to ...
3
votes
0answers
37 views

Fundamental group of 7-gon with labelling scheme $abaaab^{-1}a^{-1}$

I want to calculate the fundamental group of of a $7$-gon with labelling scheme $abaaab^{-1}a^{-1}$. I will call the quotient space $X$ with reference point $x_0$. This is what I tried: If you have ...
1
vote
1answer
32 views

Clarifying some elementary Orbit and Stabilizer questions

I have some elementary questions in learning about groups and I just want to be sure I am on the right track. Your help is greatly appreciated. Let $A = \{ \begin{pmatrix} a & b \\ 0 & ...
3
votes
1answer
52 views

Does Sylow's theorem assert the existence of subgroups of order $p^j$ for all $j=1,\dots,k$ ?

Sylow's theorem says that there exists a subgroup of order $p^k$, where $p^k$ is the highest power of $p$ dividing the order of the group. But for example if the group order is $24$, then we can ...
0
votes
2answers
44 views

How to prove the existence of this homomorphism $G_1*G_2\to H$?

Say there is a homomorphism from a group $G_1$ to another group $H$, and there is a homomorphism from $G_2$ to $H$, then there is a homomorphism from $G_1*G_2$ to $H$ induced by the former to ...
0
votes
1answer
22 views

Equality of powers of elements in a subset of a generating set for a finite group.

I'm currently reading a paper on finding Hamiltonian paths in Cayley graphs and the author makes a claim that I can't seem to understand. Let $G$ be a finite, nilpotent group, $N$ a normal subgroup ...
0
votes
1answer
46 views

Existence a group with an infinite Frattini subgroup

Let $G$ be group and $Fix(G)$ to be the set of all elements of $G$ that are fixed by all the automorphisms of $G$. Do there exist a group such that $Fix(G)$ is infinite? Thank you.
5
votes
1answer
66 views

group-like structure texts.

I was reading Dummit and Foote to be ready for my group theory text, but my teacher seems to be paying special attention to things with less structure than groups, for example monoids, semigroups, and ...
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vote
2answers
330 views

Example of an infinite group where every element except identity has order 2

Find an infinite group, in which every element g not equal identity (e) has order 2 Does this question mean this: the group that fail condition (2) which is no inverse and also that group must ...
5
votes
1answer
66 views

Maximal height of subgroups in $S_n$?

In the process of solving some exercise, I became curious about the maximum height of a chain of subgroups in $S_n$. More specifically - what is the maximum length k of a chain of subgroups $\{e\} ...
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vote
0answers
50 views

Probability that elements in a noncommutative group commute

We consider $G$, a noncommutative group which contains $n$ elements. Show the probability that $2$ elements of $G$ commute is lower than $\frac{5}{8}$.
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votes
0answers
39 views

A subgroup of functions under multiplication

Let $G$ be a group of functions from $R$ to $R^*$ under multiplication. Let $H=\{f\in G| f(1)=1\}$. Prove that $H$ is a subgroup of $G$. Can someone make the structure of $G$ more clear possibly with ...
-2
votes
1answer
53 views

Centraliser of a cyclic subgroup

Suppose $a$ belongs to a group $G$ and order of $a$ is $5$. Prove that $C(a)=C(a^3)$, where $C(a)$ is the centralizer of $a$ in $G$. We can show that $C(a)=C(a^4)$ as we always have the identity ...
2
votes
1answer
13 views

Understanding step in proof of semi-group with equations $a \ast x = b$ and $x' \ast a' = b'$ admitting solutions for any a, b, a', b' is a group

I do not understand one point in the proof from user68061 of this statement There is a similar question (here) but this time with no unicity condition for the solutions x and x'. Henning Makholm ...
2
votes
0answers
36 views

What are the main differences among representations of $GL(n, \mathbb{R})$, $GL(n, \mathbb{C})$, and $GL(n,k)$?

What are the main differences among representations of $GL(n, \mathbb{R})$, $GL(n, \mathbb{C})$, and $GL(n,k)$? Here $k$ is a non-archimedean local field. Thank you very much.
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2answers
57 views

A finite divisible group is trivial

I am having trouble seeing why a finite divisible group is necessarily trivial. Why does this have to be the case?
1
vote
1answer
47 views

Are the two inverses in the free group same?

Set G is a group, and the set C is contained in G. Set C contains only 2 elements a and b such that they are the inverse of each other.Set F(C) is the free group made by C. Then in F(C), it contains ...
1
vote
1answer
35 views

Decomposing $g=xy$ where $\left|g\right|=\left|x\right|{\cdot}\left|y\right|$

Throughout, let us assume we are working with a finite group $G$. The order of an element $g\in G$ is denoted by $\left|g\right|$. It is a standard exercise to prove that if $x, y\in G$ have ...
1
vote
1answer
32 views

Order of a group of matrices

I have to calculate the order of the group $G=\operatorname{GL}(n,\mathbb{Z}_m)$ with $m=p^k$ for $p$ prime and $k\in \mathbb{N}$. So I was thinking on the homomorphism $\det: G \rightarrow ...
2
votes
0answers
48 views

If G is direct product of nonabelian & simple groups. Then prove that Aut(Aut(G)) has only inner automorphisms.

If G is direct product of nonabelian & simple groups. Then prove that Aut(Aut(G)) has only inner automorphisms. Here we can restrict any map f belongs to Aut(Aut(G)) then restrict f to Soc(Aut(G)) ...
2
votes
1answer
24 views

Relative orders of an element with respect to a subgroup

There is a theorem in an old monograph: Theorem 1. A pair of subgroups $A$ and $B$ forms a distributive pair if and only if for every element $c$ of $A\vee B$, not in $A$ nor in $B$, its relative ...
2
votes
1answer
30 views

Criteria of regularity that a Cayley's Diagram should meet .

As referred in the Visual group theory Book by Nathan Carter- The unofficial definition of a group says that : A group is a collection of actions satisfying the rules: 1. there is a predefined list ...
4
votes
2answers
96 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 ...
4
votes
2answers
50 views

Is $<\mathbb Q^+, \times>$ the free abelian group on countably infinitely many generators?

It seems to make sense to me that it should be, with the generators being the set of primes. However, I'm not sure that my intuition is right. Additionally, would this not be contradicted by the fact ...
2
votes
1answer
24 views

Nilpotent and Invertible elements in commutative ring with 1

Let $R$ be a commutative ring with $1$, $S$ a subring also with $1$. Suppose $R\setminus S$ contains a nilpotent element. Prove that $R\setminus S$ also contains an invertible element. Attempt at ...
2
votes
1answer
46 views

what are all finite subgroups of $\mathbb{Z}^n \rtimes \mathbb{Z}_2$?

$\mathbb{Z}^n \rtimes \mathbb{Z}_2$ := $\{(u_1,u_2....,u_n,t), u_iu_j=u_ju_i, tu_jt=u_j^{-1},t^2=1\} $ what are all finite subgroups of $\mathbb{Z}^n \rtimes \mathbb{Z}_2$? (in terms of the ...
2
votes
1answer
69 views

If a finite set $G$ is closed under an associative product and both cancellation laws hold, then it is a group

Problem Suppose a finite set $G$ is closed under an associative product and that both cancellation law hold in $G$. Prove that $G$ must be a group. Also show by an example that if one just assumed ...
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0answers
23 views

A question regarding conjugacy classes of central involutions.

An involution $a$ of a group $G$ is called central if there exists a sylow $2$-subgroup $H$ of $G$ such that $a \in C_G(H)$. Clearly if an involution is central then its ever conjugate is also ...
2
votes
1answer
28 views

A question on the automorphism of simple graph with distinct eigenvalues of adjacency matrix

Let G be a graph. If G is simple(i.e no loops), and the eigenvalues of adjacency matrix A are distinct, then the automorphism of G is abelian. It seems that the automorphism from G to G itself is only ...
4
votes
2answers
100 views

Action of $G/H$ on $H_n(H;M)$

I'm currently studying group cohomology and have trouble with the Hochschild-Serre spectral sequence. My problem is this: Given a short exact sequence of groups $$ 0 \to H \to G \to G/H \to 0$$ how ...
5
votes
2answers
623 views

Show that a finite group with certain automorphism is abelian

Let $G$ be a finite group and $f:G\to G$ an isomorphism. If $f$ has no fixed points (i.e., $f(x)=x$ implies $x=e$) and if $f\circ f$ is the identity, then $G$ is abelian. (Hint: Prove that every ...
0
votes
1answer
40 views

easy short exact sequence question

Suppose I have have a short exact sequence of finitely generated Abelian groups $0 \longrightarrow G \overset{f}\longrightarrow H \overset{g}\longrightarrow K \longrightarrow 0$. Suppose I have a ...
6
votes
2answers
539 views

Prove that a group with certain property is abelian

Let $G$ be a group with the following property: If $a, b,$ and $c$ belong to $G$ and $ab=ca$,then $b=c$. Prove that $G$ is abelian. I think I have the answer for the finite $G$ case. Then, for a ...