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

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16 views

Can we extend this presentation as an infinite vector?

The motivation to this question can be found in: How I can express $(x,y)∈G$ by using the $r$ independent points $P_1,P_2,\ldots,P_r$ My question is: The group $C(ℚ)$ is a finitely generated Abelian ...
2
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2answers
88 views

Is an arbitrary group generated by a traversal of the conjugacy classes?

Let $G$ be a group, and let $\mathcal C$ be the collection of conjugacy classes of $G$. Let $S$ be a traversal of $\mathcal C$ (that is $S$ contains exactly one element from each set in $\cal C$). ...
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1answer
31 views

what are the other 2 nontrivial elements of the automorphism group of $\Bbb Z/5\Bbb Z$?

It is known that the automorphism group of the units of $\Bbb Z/5\Bbb Z$ is isomorphic to the cyclic group of order $4$, so the automorphism group must also have $4$ elements. The two nontrivial ones ...
2
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2answers
271 views

Is isomorphism not always unique?

Given two isomorphic groups G and H, is it possible that two or more functions define their isomorphism? Also, is it possible that another group say, L is isomorphic to G but not to H?
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1answer
42 views

Solvable groups and orders of elements

I am trying to prove the following result. Let $G$ be a group containing elements $x$ and $y$ such that the orders of $x$, $y$, and $xy$ are pairwise relatively prime; prove that $G$ is not ...
2
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2answers
88 views

Cyclic Group Presentation [on hold]

Show that the the group with presentation $$\langle x, y\ \mid\ x^2=y^2x^2y,\ (xy^2)^2=yx^2, \ yx^{-1}y^2=x^7\rangle $$ is cyclic of order 24.
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2answers
58 views

Free action by cyclic group.

Let $G$ be a group acting on a set $X$. If $g\in G$ has no fixed points, prove or disprove the cyclic group $\left \langle g \right \rangle$ acts freely on $X$. edit: Can also assume $g$ has finite ...
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1answer
25 views

Finite groups with a cyclic maximal subgroup.

In the book A Course in the Theory of Groups by Derek J.S. Robinson, Finite $p$-groups with a cyclic maximal subgroup are classified. Now I wish to know whether finite groups with a cyclic maximal ...
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28 views

Can the derived subgroup be realized as an intersection of stabilizers?

For any group $G$, we have that $Z(G)=\bigcap_{x\in G}C_G(x)$ and that each $C_G(x)$ is the stabilizer of $x$ when $G$ acts on itself by conjugation. Is there a similar representation for $G'$? That ...
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1answer
42 views

Finitely generated, periodic group such that each conjugacy class is finite must be finite?

This is essentially a repeat of this question. However, the OP didn't seem to put in any work towards a solution and didn't provide context. Nevertheless, I'm still interested in the solution, and I ...
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1answer
44 views

Finitely generated group 3 [duplicate]

Let $G$ be a group generated by a finite subset $X$, such that Every element of $X$ has finite order; The number of $G$-conjugates of any element of $X$ is finite. Is it true that $G$ is ...
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26 views

Number of normal subgroups of a non-abelian group of order 21 [on hold]

How many normal subgroups can a non-abelian group G of order 21 have other than the identity subgroup {e} and G? a)0 b)1 c)3 d)7
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0answers
17 views

Existence of Generalized Hadamard matrices

Does there exist a Generalized Hadamard matrix of order 20 over an Elementary abelian group of order 4 ,GH(20,EA(4))?
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1answer
46 views

A question about Tarski Monster Group

Let $\alpha$ be a cardinality. Is there a Tarski moster group with exacly $\alpha$ non-trivial proper subgroups‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌‌?
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1answer
36 views

Absolutely and Relatively free abelian groups.

I see that there are the notions of absolutely free abelian group and relativley free abelian group. Could you please explain the difference between the two notions. Thanks!!
3
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1answer
37 views

Free cyclic subgroups in a non-abelian group

Is there any non-abelian group $G$ such that for each $a\in G$ and any automorphism $g:\left<a\right>\to \left<a\right>$ the function $$f:G\to G$$ $$f(x) = \begin{cases} g(x) & \text{ ...
6
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0answers
52 views

I would like to show that all reflections in a finite reflection group $W :=\langle t_1, \ldots , t_n\rangle$ are of the form $wt_iw^{-1}.$

I would like to show that all reflections in a finite reflection group $W := \langle t_1, \ldots , t_n\rangle$ are of the form $wt_iw^{-1}$ for some $i$ and some $w \in W$ Clearly all such elements ...
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0answers
44 views

Finitely generated groups 3 [on hold]

Let $G$ be a group generated by a finite subset $X$, such that Every element of $X$ has finite order; The number of $G$-conjugates of any element of $X$ is finite. Is it true that $G$ is ...
0
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1answer
44 views

Inconsistent definition of Sylow p-subgroup

Here is the definition of a Sylow $p$-subgroup from Wikipedia: For a prime number $p$, a Sylow $p$-subgroup (sometimes $p$-Sylow subgroup) of a group $G$ is a maximal $p$-subgroup of $G$, i.e., a ...
6
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1answer
80 views

About translating subsets of $\Bbb Z.$

This is a continuation of About translating subsets of R2. Is it possible to find a pair of sets $A,B\subseteq\Bbb Z$ such that A is a union of translated (only translations are allowed) copies of ...
2
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1answer
28 views

Constructing an indicator function from a braid group which represents 'all strings have returned to their initial position'.

TL;DR Is there a well-defined closed formula from the braid group $B_n$ to $\{-1,1\} \left(\text{ or }\{0,1\}\right)$ which represents whether all the strings have returned to their initial ...
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1answer
28 views

What are the elements of Z/10Z and of Z/2Z×Z/5Z, and identify which elements correspond under the map g from Z/10Z to Z/2Z × Z/5Z. [on hold]

What are the elements of Z/10Z and of Z/2Z×Z/5Z, and identify which elements correspond under the map g from Z/10Z to Z/2Z × Z/5Z. I know the elements of Z/10Z are {1,3,7,9}, is that the same for ...
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1answer
22 views

Prove that $K=k(\alpha)$

Prove: If K\k is a Galois extension and $\alpha \in K$ with $\sigma(\alpha)\neq \alpha$ for all $\sigma \in Gal(K,k)\backslash \lbrace id_K \rbrace$, than $K=k(\alpha)$.
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3answers
104 views

Finitely-generated group such that all (non-trivial) normal subgroups have finite index implies all (non-trivial) subgroups have finite index?

Let $G$ be a finitely generated group such that every non-trivial normal subgroup has finite index. Does it follow that every non-trivial subgroup of $G$ has finite index? This question arose as ...
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2answers
66 views

We have a map $g : \mathbb{Z}/24\mathbb{Z} → \mathbb{Z}/6\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z} $. What is the kernel of $g$?

We have a map $g : \mathbb{Z}/24\mathbb{Z} → \mathbb{Z}/6\mathbb{Z} \times \mathbb{Z}/4\mathbb{Z}$ given by $g(x+24\mathbb{Z}) = (x + 6\mathbb{Z}, x + 4\mathbb{Z})$. What is the kernel of $g $? In ...
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1answer
85 views

Trivial elements in $T(a,b,c)$

Consider the group $T(a,b,2)=<x,y|x^a, y^b, (xy)^2>$ and assume none of $a$ or $b$ is equal to $2$. How can one list all the trivial words (say up to length $11$ and apart from $(xy)^{2n})$) in ...
2
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0answers
64 views

Cokernel of injective endomorphisms of a finitely generated free abelian group

By $\text{GL}^+(n,\mathbb{Z})$ we mean the set of $n×n$ invertible matrices with positive determinant and entries from $\mathbb{Z}$. For given $A \in \text{GL}^+(n,\mathbb{Z})$ let ...
2
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58 views

Groups such that all elements of even order are in $G'$

We know that $G=A_4$ is a group such that all elements of even order in $G$ are in $G':=[G,G]= V_4$, the klein four group. Are there other examples or classes of groups where all elements of even ...
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1answer
26 views

If the quotient by the $i$th center is cyclic, does it follow that the original group is abelian?

Let $G$ be a group such that there exists an $i$ such that $G/Z^i(G)$ is cyclic. Does it follow that $G$ is abelian? This question is a generalization of the well known fact that if $G/Z(G)$ is ...
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1answer
40 views

Are these exactly the abelian groups (2)?

This is a continuation of Are these exactly the abelian groups? I would like to consider another condition on a group and see if it implies commutativity. The condition is $$(\forall A,B\subseteq ...
2
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1answer
39 views

$\mathbb{F}_2 \times \mathbb{F}_2$ is not subgroup separable

I read that $\mathbb{F}_2 \times \mathbb{F}_2$ is not subgroup separable (ie. for every finitely generated subgroup $H$ and $g \notin H$, there exists a finite index subgroup $K$ such that $H \subset ...
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1answer
57 views

Bound on the index of an abelian subgroup in discrete subgroup of the euclidean group?

$\DeclareMathOperator{\isom}{Isom}$A discrete subgroup of the group of isometries in euclidean space is almost abelian. By this I mean that for each $n$ there exists $m$ such that for any discrete ...
2
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1answer
55 views

Write cyclic groups of order $p^n$ in terms of simple groups

Some say that studying simple groups helps you understand the structure of non-simple groups. How can I write in terms of simple groups $\mathbb{Z}_{p^n}$? Eg. $\mathbb{Z}_9$
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2answers
68 views

Abelianization of the free product of two cyclic groups.

Suppose that $G=G_1*G_2$ where $G_1$ and $G_2$ are cyclic of orders $m$ and $n$ respectively. Show that $G/[G,G]$ has order $mn$. Can anyone help me?
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1answer
32 views

Commuting Elements in a Free Product of Cyclic Groups

In the free group with two generators $F_2\cong\mathbb Z *\mathbb Z$ ($*$ denotes the free product), if two elements $a$ and $b$ commute, then there exists an element $w\in F_2$ such that $\langle ...
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0answers
29 views

To derive commutativeness of any group from the normality of all its subgroups & some other conditions

If a group is abelian then it is known that every subgroup of the group is normal ; is the converse true i.e. if every subgroup of a group is normal , then is it true that the grroup is abelian ? If ...
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0answers
60 views

Algebraic constructions to add a real to a sub-group of $\langle \mathbb{R}^+,.\rangle$

Consider the group $\langle \mathbb{R}^+,.\rangle$ and let $r$ be a real number (e.g. $r=\sqrt{2}$). I would like to know about all known algebraic constructions to build a sub-group of $\langle ...
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0answers
22 views

Is $U/U(w) = U \cap w U^- w^{-1}$?

Let $U$ be the maximal upper unipotent subgroup of $GL_n$ and $U^{-}$ maximal lower unipotent subgroup of $GL_n$. Let $U(w) = U \cap wUw^{-1}$. Is $U/U(w) = U \cap w U^- w^{-1}$? Thank you very much.
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1answer
86 views

(Theorem) If $G$ is a simple group of odd order , then $G \cong \mathbb Z_p$ for some prime $p$.

I am studying Dumit Foote. I have seen this result in this book. Please help me solve this. Thank you.
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1answer
41 views

Under what conditions can the symmetric group be isomorphic to the abelian group?

The symmetric group is the set of all permutations. My question addresses the representability of the symmetric group using only additions. I am guessing that on the finite field $\mathbb{Z}/n ...
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176 views

A field having an automorphism of order 2

The following fact is used in the Unitary space. If $F$ is a field having an automorphism $\alpha$ of order 2. Let $F_0=\{a\in F: \alpha(a)=a\}$. Then $|F:F_0|=2$. Is there any easy proof (or ...
6
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1answer
94 views

Elementary equivalence of free groups

This must be known inside out by model theorists by I have no cluse whether the following is true or not: Denote by $F_n$ the free group on $n$ generators. Suppose that $n\neq m$. Are the groups ...
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1answer
37 views

Subgroups of $\mathbf{Z}/20\mathbf{Z}$ using Lagrange's Theorem

According to Lagrange's Theorem, what are the possible sizes of the subgroups of $\mathbf{Z}/20\mathbf{Z}$? I have no idea how to go about answering this. I have a feeling that I should be ...
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1answer
22 views

How can i prove that a cartesian product is isomorphic to another cartesian product

$\def\<#1>{\left<#1\right>}\def\Z{\mathbb Z}\<\Z_6, \oplus> \times \<\Z_{10},\oplus>$ is isomorphic to $\<\Z_2, \oplus> \times \<\Z_{30}, \oplus>$ i know i have to ...
0
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0answers
25 views

What is M-bar in factor groups? [duplicate]

If G is a group and N $\triangleleft$ G, show that if $\bar{M}$ is a subgroup of G/N and M = {a $\in$ G | Na $\in$ $\bar{M}$}, then M is a subgroup of G, and M $\supset$ N. If $\bar{M}$ is normal in ...
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Tricky factor groups questions [closed]

If G is a group and N $\triangleleft$ G, show that if $\bar{M}$ is a subgroup of G/N and M = {a $\in$ G | Na $\in$ $\bar{M}$}, then M is a subgroup of G, and M $\supset$ N. If $\bar{M}$ is normal in ...
5
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1answer
40 views

Characters of a Group: two definitions

If $G$ is an abelian group, the characters associated to the rapresentations of $G$ over $\textrm{GL}_1(\mathbb C)=\mathbb C^\ast$ are simply the group homomorphisms: $$\chi:G\longrightarrow\mathbb ...
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1answer
34 views

A question on the intuition of decomposition of the element of symmetry group

Any element of symmetry group $S_{n}$ can be decomposed as products of transpositions. Any m-cycle can be decomposed as m-1 transposition products. How should I think of this decomposition? Is there ...
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0answers
47 views

Wick rotation from $SO(2)$ to $SO(1,1)$: Howto

Both $SO(2)$ and $SO(1,1)$ are subgroups of $SO(2,\mathbb{C})$, one can choose a 1-parameter family of subgroups of $SO(2,\mathbb{C})_{\sigma(\phi)}\subseteq SO(2,\mathbb{C})$ defined by: ...
2
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1answer
46 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} = ...