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

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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 ...
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1answer
14 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} ...
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0answers
16 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 ...
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2answers
34 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). ...
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3answers
49 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 ...
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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 $a$?
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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
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1answer
40 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?
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1answer
36 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 $G$ ...
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1answer
40 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 ...
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0answers
34 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 ...
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0answers
38 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 ...
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2answers
46 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?
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1answer
90 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 ...
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1answer
71 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.
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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 ...
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2answers
120 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.
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30 views

A proposition to check two isomorphic nonabelian finite simple groups [on hold]

Let $S$ and $T$ be two nonabelian finite simple groups and $G=S \times T$. How do we prove the following proposition: S and T are isomorphic if and only if G has a maximal proper subgroup which ...
0
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1answer
42 views

Let G be a group s.t g^3=e for all g in G. Prove that G is abelian. [duplicate]

Let G be a group s.t g^3=e for all g in G. Prove that G is abelian. From here I have got (ab)^2=(b^2)(a^2) & (ab)^3=a((ba)^2)b Then what??
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2answers
38 views

Prove that $G$ has a element whose order is least common multiple of $m$ and $n$.

Let $G$ be an abelian group and suppose that $G$ has elements of order $m$ and $n$ respectively. Prove that $G$ has an element of order $lcm[m,n]$
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0answers
25 views

Prove that $G$ must be abelian with some given conditions. [duplicate]

Let $G$ be a finite group whose order is not a multiple of $3$. Suppose that $(ab)^3=a^3b^3$ for all $a,b\in{G}$. Prove that $G$ must be abelian.
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0answers
36 views

No cycles in finite coxeter graphs

Is there an elementary (no consideration of root systems involved) proof of the fact that the graph of an finite coxeter system doesn't entail any cycle? I got as far as this: If there were any cycle ...
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3answers
87 views

Proving that the center of a factor group is trivial

Prove that the center of the factor group $G/Z(G)$ is the trivial subgroup $[e]$. So far I've proved that $Z(G)$ is a normal subgroup of $G$ for all $a \in G $.
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2answers
51 views

Prove that if G is a cyclic group with more than 2 elements, then there always exists an isomorphism ϕ:G→G that is not the identity mapping [duplicate]

The full question is Prove that if G is a cyclic group with more than 2 elements, then there always exists an isomorphism ϕ:G→G that is not the identity mapping.I have no idea where to start?
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1answer
29 views

General lists of techniques to prove whether a set is a generator of a matrix group

It seems like a rather common problem in group theory, at least in undergraduate maths, to check whether a set is a generator of a group. The question is usually as follow: Given a group $G$, and a ...
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1answer
44 views

N is a normal subgroup of G if $aNa^{-1} \subset N $ for all $a ∈ G$. Prove that in that case, $aNa^{-1} = N $.

I said if N is a normal subgroup of G when $aNa^{-1} \subset N $ aN = Na as N is a normal subgroup of $G$. Therefore $aNa^{-1} = Naa^{-1} $ and $aNa^{-1} = N $. I would like to go with this proof ...
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1answer
68 views

What are the non-trivial normal subgroups of $O(3)$?

What are the non-trivial normal subgroups of $O(3)$? My guess is that the only one is $SO(3)$, but it's really only a guess, based on the fact that $O(3)$ is disconnected 3-manifold of $SO(3)$ and the ...
2
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1answer
64 views

Structure of a group, $G$, of order $pq$ where $p, q$ are prime.

There is a proposition in Beachy and Blair's Abstract Algebra that I don't entirely follow. The proposition is the following: Let $G$ be a group of order $pq$, where $p > q$ are primes. a) If ...
2
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1answer
73 views

A certain two-step subgroup of a nilpotent group

Let $\Gamma$ be a finitely-generated, torsion-free, nilpotent group, of nilpotency class $n\ge 2$. Is there an $N \lhd \Gamma$, such that (i) $N$ is two-step nilpotent, (ii) $\Gamma / N$ is torsion ...
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2answers
79 views

Characterization of nilpotent groups

It is trivial that a group $G$ is abelian if and only if every subgroup of $G$ with two generators is abelian (i.e., any two elements commute). If $G$ is a nilpotent group, every subgroup with two ...
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0answers
31 views

Compact operators on $L^2(G)$ as a reduced cross product of $C_0(G)$ and $G$.

If any of the terminology is unclear then please don't hesitate to point it out. My question is: is it true that when $G$ is a locally compact second countable group then: \begin{equation*} C_0(G) ...
2
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1answer
32 views

Is a torsion-free discrete abelian group of finite rank isomorphic to a subgroup of $\mathbb{Q}^k_d$?

If $G$ is a torsion-free discrete abelian group of rank $k$, then is it true that $G$ is isomorphic to a group $H$, where $\mathbb{Z}^k < H < \mathbb{Q}^k_d$? Here, $\mathbb{Q}^k_d$ is the ...
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0answers
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Commutator series of some group

What mean length of commutator series of group $G$? Is it true that if this length is finite, then $G\not= G'$?
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22 views

Positive definite functions generated by irreducible representations — what do people call them?

Let $G$ be a group and $\pi:G\to B(H)$ be its irreducible unitary representation (one can endow $G$ with topology and claim that $\pi$ is continuous in some sense, this doesn't matter). For a given ...
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1answer
60 views

Projective special linear groups

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 ...
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1answer
67 views

Free product as automorphism group of graph

Let $A$ and $B$ be two groups. We define following graph $X$. The set of vertices is the left cosets $gA$ and $gB$ where $g\in A*B$ (By $A*B$, I mean the free product of $A$ and $B$). The edges of the ...
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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 ...
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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 ...
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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 ...
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2answers
84 views

On finite exponent abelian $p$-groups

Let $G$ be an abelian $p$-group non-isomorphic to any group of the form $H\times K$ where $H$ and $K$ are nontrivial groups. And let $\{|a|\mid a\in G\}$ have an upper bound in $\Bbb N$ . Is $G$ ...
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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 $H=K$ ?
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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 ...
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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 ...
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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 ...
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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 ...
2
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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 ...
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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 ...
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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.
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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 ...
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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 ...