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

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Subgroup of a Direct Product

Let $G$ and $H$ be groups and $G\times H$ their direct product. a) Prove that $\{(x,e) : x\in G\}$ is a subgroup of $G\times H$ b) Prove that $\{(x,x) : x\in G\}$ is a subgroup of $G\times G$ I ...
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126 views

Elements of order $10$ in $\Bbb Z_2 \times \Bbb Z_{10}$

How many elements in the group $\mathbb Z_2 \times \mathbb Z_{10}$ are of order $10$? I think the easiest way to answer this question might be to write them out, but I'm not sure how to write ...
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93 views

Can you give me a good alternative to Rotman's Group Theory book?

I've been trying to learn out of Rotman's book "An Introduction To The Theory of Groups" for the last few months, and it's rough going. I've been studying Chapters 7, 10, and 11 in particular, and ...
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45 views

For what n does $|A|=n, \ A \in SL_2(\mathbb{Z})$?

Could you help me check for what n does $|A|=n, \ A \in SL_2(\mathbb{Z})$? I know that given two eigenvalues $\alpha, \beta$ of a $2 \times 2$ matrix $A$, $A^n = \alpha ^n (\frac{A-\beta I}{\alpha - ...
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73 views

Can one prove $(g_1H)(g_2H) = (g_1g_2)H$, $g_1, g_2 \in G$ if and only if $gH = Hg$ for all $g \in G$?

Let $G$ be a group and $H$ a subgroup of $G$. I have proven in an exercise that $gH = Hg$ for all $g \in G$ implies $(g_1H)(g_2H) = (g_1g_2)H$, $g_1, g_2 \in G$ where $(g_1H)(g_2H) = (xy | x\in g_1H, ...
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145 views

Burnside Theorems …

I have read that Burnside's theorem implies that a group with order $p^aq^b$ cannot be simple. Well how can I prove that: From Burnside's conjugation theorem (If G has a conjugacy class whose order ...
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242 views

Groups and isomorphism

Let $G$ be the group of all nonzero complex numbers under multiplication and let $\bar{G}$ be the group of all real $2\times 2$ matrices of the form $\begin{pmatrix} a & b \\ -b & a ...
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81 views

Symmetries and morphisms of a circle

I have a question that I don't know how to approach. Here is the question: Consider the group of symmetries of a circle denoted by $D_\infty$. Let $SD_\infty$ denote the subgroup of $D_\infty$ ...
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134 views

Show that $HK=\mathbb{Z}_n^\times$

Let $p$ and $q$ be distinct prime numbers and $n=pq$. Show that $HK=\mathbb{Z}_n^\times$ for the subgroups $H=\{[x]\in\mathbb{Z}_n^\times\mid x\equiv 1\pmod{p}\}$ and $K=\{[y]\in\mathbb Z_n^\times ...
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102 views

Is the kernel of this group action the centralizer?

In Dummit and Foote, they state "... let the group $N_G(A)$ (normalizer) act on the set $A$ by conjugation. It is easy to check that the kernel of this action is the centralizer $C_G(A)$." From ...
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107 views

Show that there exists two Sylow $p$-subgroups $P$ and $Q$ such that $[P:P\cap Q] = [Q:P\cap Q] = p$

Let G be a group with the number of Sylow $p$-subgroups different from 1 mod $p^2$. Show that there exists two Sylow $p$-subgroups $P$ and $Q$ such that $[P:P\cap Q] = [Q:P\cap Q] = p$. First, ...
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179 views

Prove that a group is a quaternion group.

The representation of the Quaternion group is $$\mathrm{Q} = \langle -1,i,j,k \mid (-1)^2 = 1, \;i^2 = j^2 = k^2 = ijk = -1 \rangle.$$ Does this imply that as long as I have found a group with $4$ ...
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101 views

PM is supersolvable group

$G$ is a finite group, $G = PM$, where $P$ is a Sylow $p$-subgroup of $G$, $p$ is the largest prime dividing the order of $G$, $P$ is normal in $G$, $M$ is maximal subgroup of $G$, $M$ is ...
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104 views

Approximation for the number of involutions?

I am interested any approximation that may be available for the following expression: $$ {\left(2n\right)}!\sum_{k=0}^{n}\frac {1} {2^k \; k! \; \left(2n-2k\right)!} $$ ... which can be expressed ...
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60 views

Question based on Langrange Theorem

The question goes: Let n be a natural number. Prove that if $G$ is a group of size $n$, and $H\leq G$ is a subgroup of order $n−1$ , then $n=2$. And my solution to the question is: Let $G$ be a ...
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377 views

Decomposition of a group into disjoint double cosets

This is taken from Lang's Algebra, exercises on groups. Let $G$ be a group and $H,H'$ be subgroups of $G$. A double coset of $H,H'$ is a subset of the form $HxH'$, $x\in G$. The first question asks ...
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69 views

Understanding a proof about finite $p$-groups

I can't follow the reasoning of the author,in this proof: let $G$ be a finite $p-$group. If $H$ is a proper subgroup of G, then $H<N_G(H)$ (clearly $N_G(H)$ is the normalizer of $H$ and p is ...
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71 views

Order of the factor group by the center equals order of the group

My question is: What are the implications if the order of the factor group $G/Z(G)$ is equal to the order of the group $G$? I know that $|G/Z(G)|=|G|$ if $|Z(G)|=1$ meaning the center is trivial. ...
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53 views

How to show that two representations are equivalent?

I am reading the lecture notes. On page 14, example of $C_{c}^{\infty}(G)$. I am trying to show that the map $A$ takes $f$ to $g\mapsto f(g^{-1})$ is an invertible element of ...
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58 views

How to show that $\pi^*(g)=\chi(\det g)^{-1}$?

I am reading the lecture notes. On page 14, how could we show that $\pi^*(g)=\chi(\det g)^{-1}$? I think that $\langle \pi^*(g)\lambda, v \rangle = \langle \lambda, \chi(\det g)^{-1} v \rangle = ...
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150 views

sufficient and necessary condition for Normality of a subgroup

Question is to : Prove that a subgroup $N$ of a group $G$ is Normal iff $gNg^{-1}\subseteq N$ for all $g\in G$. But, we define $N\unlhd G$ if $gN=Ng$ i.e., $gNg^{-1}=N$. So, question should be some ...
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71 views

Generator of all congruence classes

Is it possible for $\langle a \rangle =\mathbb Z/n\mathbb Z^*$ for $a\in \mathbb Z/n\mathbb Z^*$? I think I recall hearing it is, what is the name this element takes? I remember hearing generator ...
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80 views

Proving that doesn't exist subgroup H with order 6 in $A_4$ [duplicate]

Let $G=A_4$. Prove that does not exist subgroup $H\le G$ s.t $|H|=6$. I don't know from where to start (maybe I need to prove that if so $H\triangleleft A_4 $?) Any hint will be appreciated.
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104 views

Groups in an abstract algebra

I have been thinking my brain to find three different examples for a shape S in the plain, such that its group of symmetries is infinite.Also I was asked to draw each shape clearly why its group of ...
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154 views

group homomorphism of permutation groups

Suppose I let $$\begin{align} g_{1} &= y_{1}y_{2} + y_{3}y_{4}\\ g_{2} & = y_{1}y_{3} + y_{2}y_{4}\\ g_{3} & = y_{1}y_{4} + y_{2}y_{3}.\end{align}$$ The group $S_{4}$ (permutations of the ...
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153 views

Graph (or Group) in Astronomy

Is there an application of graph theory (or group theory) in astronomy. If there is, refer me some references.
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Proving $G\cong \Bbb{Z}_p$ if $|G|$ is prime.

For a group $G$, if $|G|$ is prime, then I have to prove $G\cong \Bbb{Z}_p$. Take any element $g\in G$. As $G$ cannot have proper subgroups, and as it also has finite order, $|g|=p$. Hence, ...
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108 views

Is the torsion-free subset not always a subgroup?

It is known that the torsion subset of a group is not, in general, a subgroup. For example, consider the infinite dihedral group $\mathcal{D}_\infty = \left < x, y \mid x² = y² = 1 \right >$. ...
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51 views

upper central series, taking intersection with subgroup ($\zeta_i G \cap H \leq \zeta_i H$)

Let $\zeta_i G$ be the upper central series of $G$. Show that for 'any' subgroup $H$, we have $$\zeta_i G \cap H \leq \zeta_i H$$ where $\zeta_i H$ is the upper central series of $H$. I tried ...
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137 views

Check condition normal subgroup in these three examples

Is the subgroup H of G is a normal subgroup of G, for: $$ i)\ G = S_5, \ H = \{id, (1,2)\} $$ $$ ii) \ G = (Sym(\mathbb{N}), \circ), \ H = \{f\in Sym(\mathbb{N}) : f(0) = 0 \}$$ $$ iii) \ G = S_4, \ H ...
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72 views

Two questions about isomorphisms of groups

Let $A_1$ and $A_2$ be Abelian groups and let $C$ and $B_i$ be subgroups of $A_i$ for $i=1,2$ and $B_1 \cap B_2 = A_1 \cap A_2 = \{0\}$. Is then true that $$(A_1\bigoplus A_2) / (B_1\bigoplus B_2) \ ...
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21 views

Degree and ramification index of a natural projection

Let $\Gamma$ a subcongruence group of $\text{SL}_2(\mathbb{Z})$, $\mathbb{H}^* = \mathbb{H} \cup \{\infty\} \cup \mathbb{Q}$ and $\overline\Gamma = \Gamma / (\Gamma\cap\{\pm 1\})$. Given the natural ...
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98 views

Conjugacy Class in Symmetric Group

This question might be duplicate because of a representation theory question. I don't know representation theory enough so I didn't tried to check that section. Please notify. I heard and experienced ...
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60 views

“Criterion for $a^i = a^j$” - question about the proof

I understand everything except for the part where the author verifies whether \begin{equation}\langle a\rangle = \{e, a^1, a^2, \ldots, a^{n-1}\}.\end{equation} To me this seems completely ...
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54 views

Partitions and Orbit Sizes

If $U,V \subset S_n$ are subgroups with $S_n//U = \{id,g_2,...,g_e\}$ and $\alpha_j$ is $\frac{1}{j}$ times the number of $i\in [e]$ s.t $[V:V \cap g_i U g_i^{-1}]=j$ then $(\alpha_1,...,\alpha_e)$ is ...
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A question about primary subgroup.

I need the definition of Primary Subgroup, and some information about it.It was not found in any resource.Can you please advise me? best regards.
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Fundamental volume of quotient group

I came across this rule in my old notes, but I need an explanation to how it could possibly originate: The theorem says that for any lattice $L$ in $\mathbb{R}^n$, the order of the quotient group, ...
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141 views

A multiple choice question on algebra [closed]

Let $G$ be a finite group such that for every two arbitrary subgroup $H$ and $K$ of $G$, $H\subseteq K$ or $K\subseteq H$. Which of the following statements is true? (a) $G$ need not to be cyclic. ...
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Classification of group extensions

For hours I have been looking for " Claude Archer. Classification of group extensions. PhD thesis,Université Libre de Bruxelles, 2002 " but I found nothing . Is there any replacement for this thesis , ...
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126 views

Questions about cosets, conjugate classes etc

Some questions about subgroups, normal subgroups, conjugate classes etc, just to make sure I understand it :-) The index of a subgroup $H$ in $G$, written as $[G:H]$ is defined as the number of left ...
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prove that , every normal subgroup of this group $G$ of the form $G_I$

let $K_1 , K_2 , .. , K_n$ are non-abelian simple groups and let $G = K_1 \times K_2 \times .. \times K_n$ prove that , every normal subgroup of $G$ of the for $G_I$ for some subset $I$ of ...
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55 views

A question about Hyperabelian Groups

A group $G$ is a hyperabelian group if has a ascending normal series with abelian factors. Prove that $F(G)$ is a hyperabelian group for all group $G$, where $F(G)$ is the Fitting subgroup of the ...
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68 views

Can we get the line graph of the $3D$ cube as a Cayley graph?

Given a graph $G=(V,E)$, the line graph of $G$ is a graph $\Gamma$ whose vertices are $E$ (the edges of $G$) and in $\Gamma$, two vertices $e_1,e_2$ are connected if, as edges in $G$, they share an ...
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75 views

Eigenspace decomposition for semisimple module

Let's start with a prime $p$ and a group $\Delta$ of order prime to $p$. Let $M$ be a finite $\mathbb{F}_p[\Delta]$-module of order a power of $p$. I want to find a decomposition into eigenspaces ...
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Prove that 2 groups $Aut(D),Aut(D')$ are isomorphism

Prove that 2 groups $Aut(D),Aut(D')$ are isomorphism given that there is bijective and holomorphic function $f$ from region $D$ onto region $D'$ (in other words, $f^{-1}$ is also holomorphic) any idea ...
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110 views

$G$ soluble and Unique minimal normal subgroup

Let $G$ be a soluble group and $N$ is only minimal normal subgroup of $G$. Is this $N=C‎_{G}‎(N)$ true?
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80 views

Compute the order of the following elements in a group

Let $G = \mathbb Z_{84}$. Let $g,h \in G$, with $g = 6, h = 80$. Compute $|g|, |h|$ and $|gh^{-1}|$.
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96 views

Projective general linear groups of order 2 = projective special linear group of order 2

If $q\geq 5$ is a prime and we consider the group PGL(2,q) the projective general linear group of order 2 and the group PSL(2,q) the projective special linear group of order 2. Can I then conclude ...
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homomorphisms of infinite groups

Prove that each of the following is a homomorphism, and describe its kernel: the function $f: \mathbb{R}^*\to\mathbb{R}_{>0}$ defined by $f(x)=|x|$ My proof step: The kernel of $f$ ...
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108 views

Diamond diagram for Correspondence Theorem

This paragraph appears in Isaacs' Algebra (chapter on homomorphisms). We comment briefly on the interpretation of the Correspondence theorem in terms of lattice diagrams, at least in the case ...