A group is an algebraic structure consisting of a set of elements together with an operation that satisfies four conditions: closure, associativity, identity and invertibility. Group theory studies groups.

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Finding a 3-embedded subgroup.

I have the group of order 108 $G=(((\mathbb{Z}_3 \times \mathbb{Z}_3)\ltimes \mathbb{Z}_3)\ltimes \mathbb{Z}_2) \ltimes \mathbb{Z}_2$ obtained from an algorithm in GAP, but I need to prove that it has ...
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1answer
49 views

Product with a normal subgroup

If $H \unlhd K \unlhd G$ and $P\unlhd G$ then does necessarily $HP \unlhd KP$? I can see this is true using the correspondence theorem since $HP/P \unlhd KP/P$ I want to try direct and prove it ...
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64 views

Commensurability of two groups

If two groups $G\bigoplus \mathbb{Z}$ and $H\bigoplus \mathbb{Z}$ are commensurable. Does it imply that the groups $G$ and $H$ are commensuarble?
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1answer
26 views

Supersolvable and pronormal subgroups

Let $G$ be a finite group such that all subgroups of prime-power order are pronormal in $G$. If $M$ is a normal $p$-subgroup of $G$ then all prime-power order subgroups of $G/M$ are pronormal in $G/M$....
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49 views

Prove that $G = \{\cos x + t \sin x: x\in\mathbb{R}\}$ is an abelian group under multiplication

where $x$ is the angle Closure is easy - Since $x$ is a real number, its $\cos$ component and $\sin$ component will be a real number. Associative property - I guess it means that $(\cos x + t\sin ...
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12 views

Polar of orthogonal set invariant under group action

I just ask the following question: Set invariant under group action Furthermore, How to prove the green part Original paper: http://arxiv.org/pdf/1403.4914v1.pdf (p.1324) Let $$O(n)=...
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2answers
107 views

Making sense of the commutator

For a group $G$, the commutator of two elements is defined as $[a,b]=aba^{-1}b^{-1}$, and is usually said to measure the extent to which the elements $a$ and $b$ fail to commute. I'm having some ...
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37 views

Set invariant under group action

I am reading a paper with the following description: $O(n): \{Y\in \mathbf{R}^{n\times n}\mid Y^TY=I\}$ We say a set $V$ is $T$-invariant if $TV\subseteq V$, where $T$ is a linear transform. ...
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1answer
26 views

If $d\mid n$, element of $(\mathbb{Z}/n\mathbb{Z})^*$ of highest order is also element of $(\mathbb{Z}/d\mathbb{Z})^*$ of highest order

I am not entirely sure if the following lemma is true, but after running a code to check, for smaller values of $n$, it holds. Can someone outline the proof or give a counterexample for it. Lemma: ...
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30 views

Combinatorial property of sets

Is the following true? For every $\varepsilon>0$ there is a finite subset $W$ of $\mathbb{N}\times \mathbb{N}\times \mathbb{N}$, such that $$|p_1(W)\cap p_2(W)\cap \{p_1(x)+p_2(x):x\in W\}\cap \{...
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2answers
21 views

Proof that Möbius transformations are group under composition - finding inverse element

The task given in my textbook was to find which algebraic structure is $(X, *)$, where $X$ is set of Möbius transformations $x\rightarrow y=\frac{ax+b}{cx+d}$ in $\mathbb R$ and $*$ is composition. I ...
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1answer
57 views

How to determine the order of the group $\langle a,b,c |a^2=b^2=c^2=(ab)^2=(bc)^4=(ca)^4=1 \rangle$?

How to determine the order of the group $\langle a,b,c |a^2=b^2=c^2=(ab)^2=(bc)^4=(ca)^4=1 \rangle$ ? I have almost no idea how to go on about this . Please help . Thanks in advance
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23 views

Is the convolution algebra a *-algebra?

Let $G$ be a finite abelian group with $n$ elements. Consider the convolution algebra $C^*(G) \subset l^2(G)$, with multiplication: $$(a * b )(g) = \frac{1}{n}\sum_{x\in G} a(x)b(g-x)$$ Is there a ...
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4answers
109 views

Infinite groups with all elements of order 2?

If G is a group such that $a^2 =e$ for all $a \in G$, where $e$ is the identity element in $G$, then $G$ is finite. This question can be proved false if we can get a group of infinite order with ...
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26 views

prime-power order subgroups of a quotient group

Let $G$ be a finite group. Suppose that $M$ is normal $p$-subgroup of $G$. Then every prime-power order subgroup of $G/M$ is either a $p$-subgroup or a $q$-subgroup of the form $QM/M$ for some $q$-...
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Problem in finding a general rule for determining $o(ab)$ if $o(a)$ and $o(b)$ are known. [duplicate]

Is there any way to determine $o(ab)$ if $o(a)$ and $o(b)$ are known, where $a$, $b$ belong to a group $G$. @Derek Holt I have proved that $o(ab) = o(ba)$.Which is very easy.Let us assume that $o(ab)...
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1answer
49 views

kernel of product of group homomorphisms

Let $f,g:A \to B$ be group homomorphisms, with $B$ abelian. Then $f\cdot g$ is also a group homomorphism. What can I say about $\ker(f \cdot g)$ in terms of $\ker(f)$ and $\ker(g)$?
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1answer
41 views

Existence of surjective map / homomorphism from an infinite group onto its symmetric group

From Cayley's theorem , we know that any group $G$ can be embedded in its permutation group $S(G)$ ; I would like to ask , If $G$ be an infinite group , then does there exist a surjection from $G$ ...
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0answers
55 views

Lie algebra equivalent definition

I am reading the paper "Affine projections of polynomials" by Neeraj Kayal. I need a clarification regarding the equivalence of two definitions of lie algebra : Let $f\in\mathbb{F}[x_1,\ldots,x_n]$ ...
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1answer
149 views

Cohomology of a group of order two with coefficients in a finite abelian group of odd order

I am looking for an elementary proof that the cohomology groups in the title are trivial in the positive degrees. In more detain, let $G=\{1,s\}$ be a group of order two, and let $A$ be an abelian ...
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1answer
44 views

A block in understadning a proof of Sylow's Theorem [duplicate]

If $G$ is a finite group of order $n$ and $p$ is a prime divisor $|G|$, then according to one of Sylow's theorems $G$ has a subgroup of order $p^m$, where $p^m$ is the highest power of $p$ which is a ...
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35 views

Group action with two normal subgroups which induce same block system

So awhile back I asked this question here on stack exchange: Normal subgroup $H$ of $G$ with same orbits of action on $X$. At the time I wasn't quite sure what I was really wanting to know about ...
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2answers
89 views

group theory, cardinality of a quotient group

Let $H$ be any finite abelian group. Define $H^n=\lbrace x^n \text{ }|\text{ }x\in H\rbrace$. It is easy to see that $H^n$ is a subgroup of $H$. It is also easy to see that $H^n\subseteq H^d$ if $d\...
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1answer
27 views

Centralizer of a cyclic sylow $p$-subgroup

Let $G$ be a group where $|G|=p^nm$ and $p$ is the smallest prime dividing $|G|$. Suppose that $P \in \operatorname{Syl}_p(G)$ is cylic. Then $C_G(P)=N_G(P)$ Now I know $|N_G(P)/C_G(P)|$ must divide $...
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1answer
59 views

Simple characterization of integers among abelian groups

This is part of an early exercise in Freyd's abelian categories. Let $\mathscr{G}$ be the category of abelian groups. The group of integers is distinguished, up to isomorphism, by the facts that: ...
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2answers
50 views

If $G$ a finite $p$-group s.t $G/[G;G]$ cyclic then $G$ abelian.

Let $G$ be a finite $p$-group and $[G,G]$ its commutator sub-group, I need to show that if the group quotient $G/[G,G]$ is cyclic then $G$ is an abelian group. My attempt is to let $g\in G$ s.t $g$ ...
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Algebraic Structures that do not respect isomorphism

One of the first things a student learn in Algebra is isomorphism, and it seems many objects in algebra are defined up to isomorphism. It then comes as a mild shock (at least to me) that quotient ...
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35 views

Converse of Lagrange theorem does not hold on $S_n$ for $n\geq 4$. [closed]

I already know that converse of Lagrange theorem does not hold on $A_n$. If converse of Lagrange theorem does hold on $A_n$ , Some group of $A_n$ has all 3cycles and it is whole group. Could you ...
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4answers
158 views

Concerning Groups having the property that intersection of any two non-trivial subgroups is non-trivial

The group of rational numbers $(\mathbb Q,+)$ has an interesting property , that the intersection of any two non-trivial subgroups of this group is non-trivial . Let us call this property the " non-...
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2answers
54 views

Are the groups $\mathbb R/ \mathbb Z $ and $ \mathbb R^2 / (\mathbb Z \times \{0\} )$ isomorphic?

Is it true that as groups , $\mathbb R/ \mathbb Z \cong \mathbb R^2 / (\mathbb Z \times \{0\} )$ ? I only know that $\mathbb R \cong \mathbb R^2$ (as groups ) but I can see no way to decide whether ...
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2answers
88 views

Are the groups $\mathbb C^* \times \mathbb R^*$ and $\mathbb R^* \times \mathbb R^*$ isomorphic ?

Consider the groups $\mathbb R^* , \mathbb C^*$ under multiplication , I know that they are not isomorphic ( as one of them is divisible but the other is not ) , my question is : Are the groups $\...
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1answer
36 views

Order of automorphism divides order of element

Question: Let $a$ belong to a group $G$ and let $\left | a \right |$ be finite. Let $\phi_{a}$ be the automorphism of $G$ given by $\left ( x \right )\phi_{a}=axa^{-1}.$ Show that $\left | \...
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1answer
23 views

Why is the order of an element in a cyclic group a factor of the order of the cyclic group it is in?

If G has an element a of order k, then the group generated by a consists of {e, a, a^2, ..., a^(k-1)} which are all distinct elements of the group generated by a whose orders are a factor of k. I don'...
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29 views

Definition of $k$-transposition group

In A monster tale: a review on Borcherds’ proof of monstrous moonshine conjecture, a $k$-transposition group is defined as follows. Recall that a $k$-transposition group $G$ is one generated by a ...
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2answers
53 views

Studying $\operatorname{Hom}(G,\mathbb{Z}/2\mathbb{Z})$ where $G$ is finite.

Let $G$ be a finite order group. Then we can write $|G| = 2^n(2m+1)$ for some non-negative integers $n$ and $m$. I'm trying to show that $\operatorname{Hom}(G,\mathbb{Z}/2\mathbb{Z})$ is an abelian ...
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Doubly transitive and solvable

I wonder if there are subgroup of $S_n$ that are solvable and doubly transitive for $n\geq 5$. Can someone find an example or prove that they don't exist?
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1answer
56 views

Automorphism of a finite cylic $p$-group

Let $G$ be a finite group. Suppose that $A$ is a cylic $p$-subgroup and $A\unlhd N_G(Q)$ for some subgroup $Q$ of $G$. Let $x \in N_G(Q)$ be an $p'$-order element. Then $x$ induces an automorphism of ...
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1answer
29 views

Sylow-p-group of matrices group over finite field.

Let $F$ be a finite field of characteristic $p$ and $U$ a subgroup of $F^*$. Let $G$ be the Group of $n*n$ upper triangle matrices over $F$ with elements of $ U $ on the diagonal. Find a Sylow-p-...
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1answer
32 views

Let $G$ a group of order $n$. If $k$ divide $n$ in a certain sense, is there an element of order $k$.

Let $G$ a group of order $n$. I was wondering that if $k$ divide $n$ in a certain sense, then there an element of order $k$. I explain. Let $g$ an element of order $m$. We know by Lagrange that $m\mid ...
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Groups and rings of order $p^2$.

Up to isomorphism there are exactly two abelian groups of order $p^2$. there are exactly two groups of order $p^2$. there are exactly two commutative rings of order $p^2$. there is exactly one ...
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3answers
25 views

Show that the two maps are equivalent

Question: Let G be a group and let $g \in G$. If $z \in Z\left ( G \right )$, show that the inner automorphism induced by g is the same as the inner automorphism induced by $zg$. That is show that,...
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32 views

subnormal subgroup of a $p$-group

I could not reason with this statement of a proof I am currently looking at. Let $G$ be a finite $p$-group with $Q\leq G$ and suppose that $H$ is a cyclic subgroup of $Q$ then $H$ is subnormal in $...
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1answer
82 views

Least symmetric group having a certain Abelian group as subgroup

Given an Abelian group $G\simeq\bigoplus_{k}\mathbb Z_{p^{n_k}_{k}}$, where $p_1\leq p_2\leq ...$ are primes, how to calculate the least symmetric group $S_n$ having a subgroup isomorphic to $G$?
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1answer
47 views

How many elements does group of symmetries of this logo have?

I could only rotate but not reflect it. So is it $2$?
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1answer
23 views

Properly discontinuous group actions - Hausdorffness

I was told to prove the following: If an action is free and satisfies that each point has a neighborhood $U$ satisfying $U \cap gU=\emptyset$ except for finitely many $g\in G$, and moreover the space ...
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1answer
55 views

When is the commutator subgroup a maximal subgroup? [closed]

Let $G$ be a group , under what conditions do we have that $G/[G,G]$ is a finite group of prime order ?
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1answer
73 views

Is it true that $SL(n, \mathbb R)=<\{ABA^{-1}B^{-1} : A,B \in GL(n,\mathbb R) \}$ >? [closed]

Is it true that $\operatorname{SL}(n, \mathbb R)=\left\langle \left\{ABA^{-1}B^{-1} : A,B \in \operatorname{GL}(n,\mathbb R) \right\} \right\rangle$?
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1answer
37 views

Finitely presented subgroup of finitely presented group

If I am given a group $G$, which is finitely presented by $\langle S \mid R \rangle$, and I am given a finitely presented subgroup $H$ of $G$. Is it true that $H$ takes the form $\langle T \mid R' \...
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0answers
19 views

Let $G = \mathbb{Z}_5 \times A_5$. if $H$ be a subgroup of $G$ of order $5$, then $H$ is weakly $s$-permutably embedded in $G$.

$H$ is called $s$-permutable in $G$ if it permutes with every Sylow subgroup of $G$. $H$ is called $s$-permutably embedded in $G$ if each Sylow subgroup of $H$ is a Sylow subgroup of some $s$-...
0
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1answer
43 views

interchanges/transpositions (how to read)

I have came across this before and just again now, in the same form of which I'm struggling to understand. Although I know it's link to parity, as a perm group pi: $$ \pi = \begin{pmatrix} 0 & 1 ...