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

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9
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3answers
118 views

Number of elements of order $2$ in $S_n$

How many elements of order $2$ are there in $S_n$? Using combinatorics I arrived at this: For $n$ even ($n=2k$) there are ${n\choose2}+{n\choose 2}{n-2\choose 2}\dfrac{1}{2!}+{n\choose 2} ...
1
vote
1answer
40 views

A direct proof of $\binom{m\,p^k-1}{p^k-1}\equiv1~(\text{mod $p$})$?

In Nathan Jacobson's "Basic algebra I" the exercises 1.13.11-14 prove the following extension of (a part of) the Sylow's second theorem: If $p$ is a prime and $p^k\bigm||G|$, then the number of ...
2
votes
1answer
39 views

Proof of Cauchy's Lemma in the case that G is abelian

I want to prove Cauchy's Lemma for abelian groups: If $G$ is abelian and there exists a prime such that $p$ divides the order of $G$, then there exists a $g \in G$ such that $p=\mathrm{ord}(g)$ I am ...
3
votes
1answer
31 views

Existence of proper I.C.C. subgroup

A countable discrete group $G$ is called I.C.C.(infinite conjugacy class) if for any $e\neq g\in G$, $\#\{sgs^{-1}\mid s\in G\}=\infty$. My question is: Is it possible for a group $G$ to be ...
1
vote
1answer
25 views

Frobenius dihedral groups

How can we show by a direct group-theoretic proof that the dihedral group $D_{2n}$ is a Frobenius group iff $n$ is an odd number?
2
votes
1answer
38 views

Characteristically simple subgroup

Let $G$ be a finite group and $H$ be a minimal normal subgroup of $G$. How can we show that $H$ is a characteristically simple subgroup of $G$?
0
votes
0answers
22 views

finite simple group of Lie type

let $G$ be a finite simple group of Lie type. would someone please help me! I want to know, is the following expression true or not? " There exists a simple linear algebraic $\mathcal{G}$ over an ...
2
votes
1answer
37 views

List of groups with specific divisors

I want to find list of finite simple nonabelian groups which their orders divisor lies in specific set of primes for example the set $\{2,3,5,7,11\}$. Is there a method to do this? Would Zsigmondy's ...
1
vote
3answers
122 views

What is so special about $a*b^{ -1}$ equivalence?

This equivalent is used often in group theory. For example, using this equivalnce you prove Lagranges theroem and also this equivalence gives you the cosets and other things. This equivalence also ...
2
votes
1answer
63 views

A homomorphism of group $G$ to be an automorphism

Let $G$ be a finite group such that $G=\langle x_{1},...,x_{t}| r_{1}=r'_{1},...,r_{k}=r'_{k}\rangle$. Now we define the homomorphism $\alpha$ of $G$ given by $\alpha({x_{i}})=y_{i}$ for any $i$ such ...
3
votes
1answer
46 views

Minimal normal subgroup in a finite group

Let $G$ be a finite group and $G=HK$ such that $H<G$. How can we show that if $K$ is an abelian minimal normal subgroup of $G$, then $H$ is a maximal subgroup of $G$ and $H\cap K=1$?
1
vote
1answer
48 views

Does $G$ is nilpotent imply so is $G/Z(G)$?

If $G$ is nilpotent then is $G/Z(G)$ also nilpotent? If so, how can I prove it? I know the definition of nilpotent group that the upper central series of $G$ goes to $G$ in the finite length.
5
votes
0answers
39 views

How to “see at a glance” the solution to the exercise “Show that $\langle (1,2,3… n),(1,2,3… m)\rangle$ contains a 3 cycle, if $1 < n < m$”?

I tried the commutator of the generators and it worked, but I had no real justification for making that computation. Is there a perspective from which trying the commutator is the "obvious" thing to ...
3
votes
2answers
50 views

Group theory exercise - verification?

I'm self-studying abstract algebra, and this is the first non-trivial group theory exercise I've done. Although it's a well-known result, I'd like to make sure it is correct as it took a good few ...
2
votes
1answer
49 views

About a nested sequence of subsets of integers

Let $(H_n)$ be a sequence of nonempty subsets of $\Bbb Z$ such that always $\{a-b\mid a,b\in H_{n+1}\}\subsetneqq H_n$. Can we deduce that there is some $n$ such that $\{a-b\mid a,b\in H_{n}\} = ...
3
votes
1answer
45 views

Find $o(b)$ if $aba^{-1}=b^2$ and given that $a^5=e$

If in a group $G$, $a^5=e$, $aba^{-1}=b^2$ for some $a,b\in{G}$. Find $o(b)$. I wrote $aba^{-1}=b^2$ as $ab=b^2a$. Then $(ab)^5=(b^2a)^5$ but then I am stucked up.
0
votes
0answers
39 views

A question from Topics in algebra-IN Herstein. [duplicate]

Prove that if a abelian group has elements of order $m$ and $n$ then it has a subgroup of order equals to $lcm[m,n]$. I am new to group theory so please explain....
1
vote
1answer
37 views

On group that has no non-abelian subgroup of order 6

Do there exits a group $G$ such that $G$ has not a non-abelian subgroup of order 6, but $Inn(G)$ is isomorphic to a non abelian group of order 6? Thank you We know that $\dfrac{G}{Z(G)}\cong ...
1
vote
1answer
46 views

Let $p$ be a prime number and let $Z=\{z\in \mathbb{C}\: z^{p^{n}}=1$ for some $n \in N\}$.

Let $p$ be a prime number and let $Z=\{z\in \mathbb{C}: z^{p^{n}}=1$ for some $n \in\Bbb N\}$. (a) Show that every proper subgroup of $Z$ is of the form $H_{k}$ for some $k$, where $H_k=\{z\in C: ...
2
votes
1answer
49 views

Is there any efficient algorithm for finding subgroups of a given finite group?

I am implementing an algorithm which finds every subgroup of given group. Here's my algorithm. Let $G$ be a group of order $n$ with elements $g_1,\cdots,g_n$. Then I consider each $\langle ...
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 ...
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 $x\in G$?
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 ...
2
votes
1answer
28 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
168 views

How can I find $\mathbb Z_4$ as an extension of $\mathbb Z_2$ by $\mathbb Z_2$?

Let $H=\{1,h\}$ and $A=\{0,a\}$ be groups, and $\pi:H\rightarrow \text{Aut}(A)$ be the trivial homomorphism. I have found $FS(H,A,\pi)=\{f_0,f_1\}$ and $IFS(H,A,\pi)={f_0}$ where ...
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 ...
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 ...
1
vote
4answers
47 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 ...
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
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
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 ...
5
votes
1answer
163 views

Why do we care about simple groups rather than indecomposable groups?

A first thing that we do to analyze something is "divide and conqur." So it is natural to consider simple groups. In the same way, it appears to be natural to consider indecomposable groups. However, ...
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 ...
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 & ...
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 ...
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
1answer
58 views

Proving that the intersection of a Sylow p-group with a normal subgroup is also a Sylow p-group

An exercise from Dummit and Foote pg. $101$ ex. $9$ asks to show the following: Let $G$ be a group of order $p^{a}n$ where $p$ does not divide $n$ and let $N\unlhd G$ so that $|N|=p^{b}m$ where ...
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
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 $\frac{5}{8}$.
0
votes
1answer
45 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.
1
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1answer
83 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
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.
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\} ...
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
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 ...