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

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2
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1answer
73 views

Projective special linear groups are co-hopfian?

Is it known if $PSL(2,\, F)$, with $F$ a field of prime $p$ characteristic (maybe with all proper subfields of finite order), is co-hopfian? I've searched everywhere but found nothing. Definition A ...
3
votes
3answers
52 views

Why $f\colon \mathbb{Z}_n^\times \to \mathbb{Z}_m^\times$ is surjective?

If $m|n$. Why the map $f\colon \mathbb{Z}_n^\times \to \mathbb{Z}_m^\times$ given by $a \mod{n}\mapsto a \mod m$ is a surjective homomorphism of groups? Attempt: I proved it is well a well defined ...
4
votes
1answer
31 views

Unique normal subgroups of every possible order

Can one characterize groups $G$ for which there is a unique normal subgroup of order $d$ for every divisor of the order of $G$? For example must they be solvable?
4
votes
4answers
124 views

If a group contains a subgroup with the order of each of its divisors, is it abelian?

Let $G$ be a group that has a subgroup of size $d$ for every $d$ that divides $|G|$. Must $G$ be abelian? It can be shown using complete induction that the converse of the above statement is true, ...
3
votes
1answer
32 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 ...
2
votes
2answers
57 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.
1
vote
2answers
66 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
1answer
89 views

Simple subgroup proof, would love some advice

Let $\mathrm{GL}_2(\mathbb R)$ be the group of invertible $2\times 2$ matrices with real entries. Consider $H$ which is the subset of matrices of the form $$A=\begin{bmatrix}1 & b \\ 0 ...
5
votes
5answers
701 views

If G is a finite group with an even number of elements, then binary product of two distinct elements is identity.

Let $G$ be finite group, which has an even number of elements. Show that at least for two (distinct) elements $g,h$ of group $G$ one has $g*g = e$ and $h*h = e$. I just started learning algebra and I ...
1
vote
1answer
61 views

Do those 'groups' have a name?

Is there a name for 'groups' that only have a neutral element on the right and an inverse for each element on the right ? If there is, does that name also hold for a neutral elt on the right and an ...
85
votes
2answers
3k views

Does a four-variable analog of the Hall-Witt identity exist?

Lately I have been thinking about commutator formulas, sparked by rereading the following paragraph in Isaacs (p.125): An amazing commutator formula is the Hall-Witt identity: ...
2
votes
0answers
32 views

a finite $p$-group of class two such that $Z(G)$ is not subgroup of $\Phi(G)$

Let $G$ be a finite $p$-group of class two such that $Z(G)$ is not subgroup of $\Phi(G)$. Also let $T$ be the minimal set of generators of $G$. Then do there exists a finite $p$-group $G$ of class ...
12
votes
1answer
321 views

What is the mathematical significance of Penrose tiles?

I have a very limited understanding of groups and symmetry gained mostly from online videos (for eg. this one), so forgive me if this sounds ignorant. Particular parts of Penrose tilings exhibit ...
0
votes
0answers
31 views

on special type of capable group

let $G$ be a group with this property: $G$ is a finite p-group of class two. $G=\langle x, y, Z(G)\rangle$, $|x|=|y|=p^n$ , $Z(G)$ is not cyclic, $Z(G)$ is not subgroup of $\Phi(G)$, Frattini ...
3
votes
1answer
21 views

Conjugation in a groupoid

In a group $G$ , the conjugation of any element $g \in G$ by $h$ defines an inner automorphism $\psi_h : g \to hgh^{-1}$. If one considers instead a connected groupoid $\mathcal{G}$, is there a ...
1
vote
2answers
34 views

Normal subgroup of a group

Let $H$ be a subgroup of $G$ and $K$ be a normal Subgroup of $G$. I need to prove that $KH$ is a subgroup of G where, $KH=\{\text{$kh$ : $k \in K \wedge h \in H$}\}$. Can somebody please help me in ...
2
votes
2answers
58 views

Number of conjugacy classes in finite groups

Let $G$ be a finite group. Let $C_1,C_2,\dots,C_k$ be its conjugacy classes. We denote by $C_{j\ '}=\{g^{-1}|\ g\in C_j\}$ the conjugacy class inverse to $C_j$. Set $$a_{rst} = ...
5
votes
0answers
119 views

Factorization of a finite group by two subsets

I want to write a GAP program for checking the following question. Let $G$ be a given finite group with order $n$. Is it true that for every factorization $n=ab$ there exist subsets $A$ and $B$ ...
3
votes
1answer
40 views

Conjugate Groups

$H$ and $K$ are conjugates of a group $G$ with $a \in G$, where $aHa^-1= \{aha^{-1} : h \in H\}= K$. Prove that the set $A= \{a \in G : aHa^{-1}=H\}$ is a subgroup of $G$. For $a, b \in A$, ...
2
votes
3answers
44 views

Define a normal subgroup of G

$N$ is a normal subgroup of $G$ if $aNa^{-1}$ is a subset of $N$ for all elements $a $ contained in $G$. Assume, $aNa^{-1} = \{ana^{-1}|n \in N\}$. Prove that in that case $aNa^{-1}= N.$ If $x$ is ...
0
votes
0answers
52 views

Understanding a proof about splitting of short exact sequences.

I am reading a paper by Keith Conrad about the splitting of exact sequences. I have a few questions about one particular section. This is Theorem 3.3 in this paper ...
1
vote
2answers
39 views

How to describe $G/U$?

Let $G=SL_2(\mathbb{C})$ and let $U = \{\left( \begin{matrix} 1 & x \\ 0 & 1 \end{matrix} \right): x \in \mathbb{C}\}$. We have an action of $U$ on $G$ by right multiplication. By definition, ...
-1
votes
1answer
93 views

A group with a finitely generated subgroup of finite index

$G$ is a group, and $H$ is a subgroup of $G$ with index $[G:H]=n$. Prove or disprove the following: If $H$ is a finitely generated group then G is a finitely-generated group. If $a\in G$ ...
0
votes
2answers
57 views

Find a group G such that Z(G/Z(G)) is nontrivial when Z(G) is the center of G.

So far I've tried using the quaternion group : $G = \{-1, 1, i, -i, j, -j, k, -k\}$ and $Z(G) = \{-1, 1\}$. I'm kind of stuck from here?
0
votes
1answer
28 views

Homomorphic image of intersection equals intersection of homomorphic images?

This is a tangent of this question. I wanted to remark in my answer that it is not generally true that given a group homomorphism $f:G\rightarrow H$ and two subgroups $X,Y\leq G$ that $$f(X\cap ...
1
vote
1answer
27 views

Proving $H$ is a normal subgroup of $G$ if $H$ and $gH$ ($g\notin H$) are the only distinct left cosets

Let $G$ be a group, and $H$ a subgroup of $G$. If $H$ and $gH$ where $g\notin H$are the only two distinct left cosets in $G$, prove that $H$ is a normal subgroup. I understand that $\{H, gH\}$ forms ...
1
vote
1answer
17 views

Conjugation of element lying in product of 3 groups lies in product of two groups.

I'm reading an article about tree automorphisms and I've got a problem whith something. Here it is: if $w \in \langle \gamma ^{h_1} \rangle \langle \gamma ^{h_2} \rangle \langle \gamma ^{h_1} ...
1
vote
0answers
25 views

Geometric definition of the stable commutator length

In his book, D.Calegari proves the equivalence of the algebraic and geometric definitions of stable commutator length (Proposition 2.10, p. 15). I actually have some difficulties in understanding the ...
2
votes
2answers
158 views

$Z(G)$ is not maximal subgroup of G

How to prove that $Z(G)$ is not a maximal subgroup of G, where G is an arbitrary group? Thanks in advance.
1
vote
2answers
44 views

Examples of continuous non-transitive group actions

In studying topology, I encountered this problem: Let $S$ be a topological space and let $G$ be a topological group acting continuously on $S$ (group action as $G \times S \to S$ map is continuous). ...
4
votes
3answers
196 views

Understanding presentations of groups

I'm trying to a build a better understanding of presentations. I get that a a group has a presentation $\langle S \mid R \rangle$ if it is the "freest" group subject to the relations $R$. But, for ...
8
votes
1answer
482 views

Automorphisms of non-abelian groups of order 27

What are the automorphism groups of non-abelian groups of order 27? (there are two non-abelian groups of order 27).
4
votes
3answers
59 views

Intuitive explanation of even/odd permutation

Given a permutation it can be classified as either even or odd depending on whether it is expressible as a product of even or odd number of transpositions. Is there some geometrical or intuitive ...
-4
votes
1answer
676 views

How to find right coset

$G=(Z_{20},+_{20})$ and $H=\langle [4]\rangle$. List the distinct right coset of $G$. What is order of $H+[6]$? Is $\frac{G}{H}$ isomorphic to $Z_4$? How to find right coset? Thanks in advance.
-4
votes
1answer
435 views

A multiple choice question on finite group [on hold]

Let $G$ be a finite group such that $Z(G)=1$. Let there exist $m$ such that $G$ has a unique element of order $m$. Which of the following statements is true? (a) $m=1$ (b) $m$ is prime (c) ...
3
votes
2answers
130 views

The definition of the right regular representation

I'm having difficulties understanding the definition of the right regular representation as it appears in Dummit & Foote's Abstract Algebra text. On page 132 it says Let $\pi:G \to S_G$ be the ...
1
vote
2answers
62 views

If a group has the element $a^3$ than will it have the element $a$? [on hold]

Is it necessary that in a group $G$ has a element $a^3$ than it will have the element $a$?
6
votes
1answer
93 views

$A\oplus C \cong B \oplus C$. Is $A \cong B$ when $C$ is finite, A and B infinite.

So my question is simply that for groups $A, B, C,$ if C is finite, A and B infinite and $A\oplus C \cong B \oplus C$, is $A \cong B$? My gut tells me this must be the case, and logically I can find ...
0
votes
0answers
39 views

On $O^{p'}(G)$ for a finite group $G$ (is my proof correct?)

Let $G$ be a finite group and let $N\unlhd G$. Consider $O^{p'}(G)$ which is the smallest normal subgroup of $G$ with factor group order coprime to $p$. Is $O^{p'}(G/N) = O^{p'}(G)/N$ if $N\le ...
4
votes
3answers
82 views

Finding kernel of a particular homomorphism

Let $f:\mathbb Z \to S_8$ be a homomorphism such that $f(1)=(1426)(257)$ , then how to compute $\ker(f)$ and $f(20)$ ? I know that $f(n)=f^n(1)$ but this seems too tedious ; please help
0
votes
1answer
40 views

The number of subgroups conjugate to a given subgroup of a finite group

Let $H$ and $K$ are subgroups of $G$ conjugate to each other. $A$ is defined as $$A = \{a \in G \mid aHa^{-1} = H \}$$ for all $a\in G$. Prove that $A$ is a subgroup of $G$ and prove that if $G$ ...
0
votes
1answer
45 views

Find a subgroup of $S_4$ that is isomorphic to V, the Klein group.

So I know that the Klein group is the group with 4 elements that is not cyclic but I'm stuck from there onwards?
1
vote
0answers
42 views

how is the jordan-holder theorem used in conjunction with short exact sequences to construct groups of certain order?

I am an undergraduate and have been asked to explain how simple groups can be used to construct groups of finite order. I started with reading about the extension problem in group theory and from ...
3
votes
3answers
140 views

A persisting element in all subgroups.

Let $G$ be a finitely generated abelian group and $a$ be a nontrival element of $G$ contained in all nontrivial subgroups of $G$. Is $G$ necessarily cyclic?
0
votes
1answer
41 views

2 question on solvable group's property

A theorem says a finite group G is solvable if and only if for every divisor n of $\vert G\vert$ such that (n,$\frac{\vert G\vert}{n})=1$,G has a subgroup of order n. Does this imply that G must ...
6
votes
1answer
224 views

Galois and solvable primitive permutation groups

I posted on this subject recently, but there was a misunderstanding on my side. Since my French is not very good, I misread the Galois's original paper. So let me explain my question again. Let ...
3
votes
2answers
57 views

The semidirect product as a deformation of the direct product

The way I think of the semidirect product is as a "deformation" of the direct product. Is there a way of making this intuition precise? Perhaps using some certain (co-) homology theory of groups?
9
votes
2answers
124 views

Groups of order $n^2$ that have no subgroup of order $n$

For which $n$ is there a group of order $n^2$ without a subgroup of order $n$. Such groups can not be nilpotent. This question is related to Sudokus as composition tables of finite groups.
0
votes
1answer
75 views

How to prove that every nonabelian group of odd order is not simple?

How to prove or disprove that every nonabelian group of odd order is not simple? I have no ideas concerning that.
3
votes
0answers
38 views

Groups of order $n^2$ with no subgroup of order $n$ [duplicate]

Is it possible to classify those groups whose order is $n^2$ for some natural number $n$ but which do not have any subgroups of order $n$? To be a bit more specific (in case a full classification is ...