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

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Symmetries of a Polynomial

I was wondering how many symmetries the polynomial $(x_1-x_2)(x_2-x_3)(x_1-x_3)$ has, and what they are. I got four: (i) $(x_1-x_2)(x_2-x_3)(x_1-x_3)$ (ii) $(x_2-x_1)(x_1-x_3)(x_2-x_3)$ (iii) ...
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25 views

Homomorphic images of a group [on hold]

If we consider $Q_8$ i.e. the Quaternion Group,then how to find the homomorphic images of this group?
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Using Lattice Isomorphism Theorem

I am working on this for my algebra class and I am stuck at the very end. $\textbf{QUESTION:}$ Let $p$ be a prime and let $G$ be a group of order $p^\alpha$. Prove that $G$ has a subgroup of order ...
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ring morphism from a group ring to another ring

I've read that if $S$ is a commutative ring, then $Hom_R(R[G],S)=Hom_R(R,S)\times Hom_{Gr}(G,\mathcal U(S))$. I've tried to show this equality but I couldn't. If $\phi: R[G] \to S$ is a ring ...
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How should I calculate the cosets of a subgroup of $\mathbb Z\times \mathbb Z?$

I'm trying to find the factor group $\mathbb Z^2/H,$ where $H = \{(5k,3k):k\in\mathbb Z \}.$ Would the coset of $H$ containing $(a,b)$ simply be $\{(5k + a, 3k+b):k\in \mathbb Z\}?$ If so, then how ...
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direct product of three square matrix

Suppose that $I_1$ is a $n_1\times n_1$ identity matrix and $I_2$ is a $n_2\times n_2$ identity matrix, and $H$ is $n\times n$ matrix. If $$ \bar H=I_1\otimes H \otimes I_2, $$ and we regard all the ...
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Is there a classification for the generating sets of symmetric group?

Is there a classification for the generating sets of symmetric group? Or, is there an algorithm for checking wheather a subset is a generating set? For example, can $S_7$ be generated by all its ...
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question on subgroup of compact group

Suppose $G$ is a compact metrizable abelian group, is it true that $G$ has no finite index subgroups iff $G$ is connected? Any reference or help is appreciated. Thanks in advance! Here are my ...
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Transitive action on Poincare upper half plane

I am trying to prove that the action of $SL_2(\mathbb{R})$ on $\mathbb{H}$ via $$ \left( \begin{array}{ c c } a & b \\ c & d \end{array} \right)z\rightarrow \frac{az+b}{cz+d} $$ ...
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36 views

No simple groups of order 9555: proof

While reading through crazyproject, I came across the following proof that there were no simple groups of order $9555$. However, I do not understand the step that says: "Moreover, since 7 does not ...
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Show that $D_n$ is a subgroup of $\mathbb{C}$!

For $ n \in \mathbb{N}$ and $0 \le r <n$, define $f_r : \mathbb{C} \to \mathbb{C}$ ; $z \mapsto ze^{2\pi i r/n}$ and $c: \mathbb{C} \to \mathbb{C}$ ; z \mapsto z conjugate$. a) Let $D_n = \{ f_0, ...
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57 views

Elementary group theory

Let $H$ , $K$ 2 subgroups of $G$. Prove that the union of H and K is a subgroup if and only if H is contained in K or K is contained in H.
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Lattice representation of the Klein bottle

I'm looking at the space $\mathbb{R^2}/G$ where $G = \mathbb{Z^2}$ acts by $(n,m)(x,y) = ((-1)^mx+m,y+n))$ and I'm trying to show that this is a smooth surface. I am having a couple of problems. To ...
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How can I prove that the inverse of $n-1$ in $U(n) = \mathbb{Z}_n^{\times}$ is $n-1$?

Where $U(n)$ is multiplicative group $mod(n)$. It seems obvious but how can I actually prove it? From modular arithmetics we have: $(n-1)a = nk+1$, so $a=(nk+1)/(n-1)$, which should be an integer ...
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34 views

Group of order 36

I follow a proof which show that any group of order $36$ has a normal Sylow subgroup. The author suppose that $G$ has no normal subgroups of order $9$ or $4$. Then $G$ won't have subgroups of order ...
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A question about a intransitive group [on hold]

Assume that the intransitive group $G$ has degree $n$ and minimal degree $n-1$. If no transitive constituent of $G$ has degree $1$, then they all are faithful and all except one are regular. Any ...
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Find number of elements of order p in a group

Given a group $\mathbb{Z}_{p^2q}$ where $p$ and $q$ are distinct primes, how to find the number of elements of order $p$; and how to be sure whether they exist . ($\mathbb{Z}_{p^2.q}$ is the addition ...
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24 views

Translation of an old proof

I have an old paper, Frobenius, G. (1902). Uber primitive Gruppen des Grades n und der Klasse n - 1. S. B. Akad. Berlin 1902, 455-459. in the Germany language. Is their a way to access a translation ...
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Does every homogeneous space allow a group structure?

Let $(X,\tau)$ be a homogeneous space, that is for all $x,y \in X$ there is a homeomorphism $\varphi:X\to X$ such that $\varphi(x) = y$. Is there a group operation $*:X\times X\to X$ such that ...
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25 views

Is it true that all proper normal subgroups of $D_{24}$ abelian?

Is it true that all proper normal subgroups of $D_{24}$ abelian ? If Yes, is it true only for $D_{4n}$ groups, or for all $D_{2n}$. I was trying to list all proper normal subgroups of $D_{24}$, Using ...
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125 views

How can I have a copy of this old paper? [on hold]

How can I have a copy of this old paper and a translation of it? Frobenius, G. (1902). Uber primitive Gruppen des Grades n und der Klasse n - 1. S. B. Akad. Berlin 1902, 455-459.
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Normalizer and centralizer are equivalent when $p$ is the smallest prime dividing $|G|$

Let $p$ be the smallest prime dividing $|G|$, and suppose that some $P \in \mathsf{Syl}_p(G)$ is cyclic. Prove that $N_G(P) = C_G(P)$. So I let $G=p^\alpha m$ $p$ does not divide $m$. P is cyclic, ...
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Clarification on proof that all groups of order $< 60$ are solvable

I've manged to prove that all groups of order $< 60$ are solvable, using Burnside's theorem. However, I found an alternate proof here Question about solvable groups It states that: "Note that ...
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21 views

Number of mutually non isomorphic Abelian groups

Let p and q be distinct primes. How many mutually non-isomorphic Abelian groups are there of order p^2q^4. I think there are 6 of them: p^2q^4 q, qp, q^2p q^2, q^2p^2 p, pq^3 pq, pq^3 q, q^3p^2 in ...
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46 views

The behavior of quotient groups under homomorphisms

We're learning normal subgroups, kernels, homomorphisms and isomorphisms in abstract algebra right now. I'm trying to tie the ends together: I know that if $G$ is a group, $N$ a normal subgroup of ...
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1answer
38 views

Groups of order $2\cdot 31\cdot 61$.

What are all groups (up to isomorphism) of order $2\cdot 31\cdot 61$? Letting $n_p$ be the number of Sylow $p$-subgroups of such a group, $G$, you can show $n_{31}=1$ using the Sylow theorems ...
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Epimorphism that is not surjective in the category of Torsion Free Abelian Groups

In reading about cokernels (relating to a homework question I have) I came across the following: https://www.dpmms.cam.ac.uk/~jg352/pdf/CTSheet4-2013.pdf I specifically wondered about question 5a. ...
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$n_p$ - the largest power of the prime $p$ which divides $n$

I was reading this article called "On A Theorem of Frobenius: Solutions to $x^n=1$ in Finite Groups" by I.M. Isaacs and G.R. Robinson (www.jstor.org/stable/2324902). In the third para of the first ...
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$\phi: G\to G$ is a homomorphism. Denote $Ker \phi := K$ and $ Im\phi := H$. If $K \cap H =\{e\}$, can we say that $G=KH$?

While solving a problem, I came across the following question : Let $G$ be an abelian group. Suppose $\phi: G\to G$ be a homomorphism. Denote $Ker \phi := K$ and $ Im\phi := H$. If $K \cap H ...
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1answer
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Prove $f(x) = x * a$ is bijective (preferably using inverse)

I have this question that I am stuck at. Let $(G; * ; I)$ be a group and let 'a' in $G$. Let $f : G \rightarrow G$ be the function defined by $f(x) = x * a$ for all $x$ in $G$. Prove that $f$ is ...
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understanding a quotient group

Let $G=\mathbb{Z}\times \mathbb{Z}$ . Let $K$ be the subgroup of $G$ generated by $(3,6)$ and $(3,1)$. Describe the rank and invariant factors of the abelian groups $K$ and $G/K$. My Try: Since $\phi ...
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Fermat's Little Theorem: group and multiplication modulo

$p$ is a prime number. $G$ is a group of integers $\{1,2,\dots,p-1\}$ under multiplication mod $p$. $d$ is a divisor of $(p-1)$ Is it possible to prove that the number of elements $a$ in $G$ such ...
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Find this example

Let $H=\{e,(13)\}$ be a subgroup of $S_3$. Find element $a,b \in S_3$ where $bh_2ah_1 \in aH$ but $bH\ne H$. $h_1$ and $h_2$ are elements in $H$. My friend thinks that it is (123) and (132), but I ...
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29 views

Projective representaions of $(\mathbb{Z}/3\mathbb{Z})^2$

I have a very short question: is there a faithful projective representaion $\rho: \mathbb{Z}/3\mathbb{Z}\times \mathbb{Z}/3\mathbb{Z}\to {\rm PGL}(4,\mathbb R)$? Thanks!
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38 views

Showing that f restricts to a group homomorphism

I have two abelian groups $C$ and $C'$ with corresponding homomorphisms $d:C→C$ such that $d^2=0$ and $d':C'→C'$ such that $(d')^2=0$. Then let $f:C→C'$ be a group homomorphism such that $fd=d'f$. I ...
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1answer
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Schur's Lemma: Is the isormorphism between two irreducible spaces unique?

Suppose $V_1 \neq V_2$ are two irreducible representations of the finite group G. Then Schur's Lemma says that any G-invariant map between them is either 0 or an Isormorphism. I understand that if ...
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2answers
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Question on order of elements in groups (subgroups)

I am a bit confused at the moment, but what can we say about the order of all elements in a finite (sub)group? Suppose we have a group $G$ such that $|G|=p^k$ for a prime $p$. Next let $H$ be a ...
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Group theory question for cyclic group

I met some problem during googling. The problem and its solution are next. and I'm wondering about 2nd YELLOW BOX $$ $$ $$ $$ Why $G$ has a unique element of order 2 in case of $H=G$ ? $$ $$ ...
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Number theory / Group theory: consecutive integers divisible by at least n prime numbers

Claim: There exist 15,251 successive positive integers $a_1, a_2\dots,a_{15251}$ such that each $a_i$ where ($1\le i\le 15251$) is divisible by at least 251 different prime numbers Is there a neat ...
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Group theory: subset of a finite group

Given $G$ be a finite group $X$ is a subset of group $G$ $|X| > \frac{|G|}{2}$ I noticed that any element in $G$ can be expressed as the product of 2 elements in $X$. Is there a valid way to ...
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19 views

Splitting a short exact sequence of orthogonal groups

How does one split the short exact sequence $$1 \rightarrow SO_n(\mathbb{R}) \rightarrow O_n(\mathbb{R}) \rightarrow \{\pm 1\} \rightarrow 1$$ ? I understand that there needs to be an injective ...
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Commutator subgroup of rank-2 free group is not finitely generated.

I'm having trouble with this exercise: Let $G$ be the free group generated by $a$ and $b$. Prove that the commutator subgroup $G'$ is not finitely generated. I found a suggestion that says to ...
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Let G be a group and let x be a fixed element of G. Define Γ(x) = {g ∈ G : gx = xg} [on hold]

(a) Prove that Γ(x) is a subgroup of G. (b) Let G = A4, let x = (1 3)(2 4) and let y = (2 4 3). Find (i) Γ(x) and (ii) Γ(y).
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Permutation modules and their vector space dimensions

I'm given a field $k$, a finite group $G$ and a set $S$ which $G$ acts on transitively. I'm then told to consider the permutation module $M = kS$. My first problem is understanding what the ...
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Stochastic processes on group-valued variables

I have had this question in my head for a long time, and if I don't find out the answer I may explode. So I'm familiar with a basic Ito process, let's say: $dX_t = \mu d t + \sigma d Z_t$. There ...
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Exponentials of Representations of Lie Algebras

Assume G is a lie group and g is its lie algebra. Consider a representation of G : D:G->End(V). Then there is a corresponding representation of g : d:g->End(V). My question is, when you can express ...
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Part of simple proof of nontrivial center in p-group

I'm trying to understand the proof of a Burnside theorem (as stated in Beachy's Abstract Algebra p. 328): Let $p$ be prime number. The center of any $p$-group is nontrivial. Now, In the proof they ...
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Multiplying Cosets

1) Let $ah$ be a coset of the subgroup $H$. Suppose there are two elements $ah_1\in aH$ and $ah_2\in aH$ such that $(ah_1)(ah_2)\in aH.$ Show that this implies that $a \in H$ and so $aH=H$. 2) ...
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How to define binary operation on arbitrary set in order to create a group structure.

Is it (and if yes how?) possible to define an an binary operation $*$ for an arbitrary set $M$ such that $(M,*)$ is a group? If $M$ is finite or countable infinite this is trivial, but is it also ...
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1answer
26 views

Elements of $\operatorname{SL}_2(\mathbb F_{p^n})$ of order $p^k$

Let $p > 2$ be a prime number and $n\ge 1$ an integer, and consider the group $G = \operatorname{SL}_2(\mathbb F_{p^n})$ of order $p^n(p^{2n} - 1)$. Let us denote by $\operatorname{Inn}(G)$ (the ...