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

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Do you know this notation in group theory?

Somebody know this notation in group theory: $$X^G,$$ where $G$ is a group and $X$ aparently is a subset of G? I've come across with this notations in the following problem: Show that $X^G = ...
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46 views

Is linear space totally different from group?

Linear space builds on Abelian group. My question is, is linear space TOTALLY different from group? Is it true that some properties of linear space are the properties of the Abelian group? Actually, ...
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199 views

Transvection matrices generate $SL_n(\mathbb{R})$

I need to prove that the transvection matrices generate the special linear group. I want to proceed using induction on n. I was able to prove the 2x2 case, but I am having difficulty with the n+1 ...
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69 views

About the quotient group of degree zero divisors on $C$ by the principal divisors on $C$

Let $C$ be an elliptic curve with distinguished point $O$. My question is about a mathematical desription of this set denoted by $Pic(C)$ which is the quotient group of degree zero divisors on $C$ by ...
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53 views

$G=C_{p}\times C_{p^2}$, describe $\mathrm{End}(G)$

I was asked to describe the group of the endomorphism of $G=C_{p} \times C_{p^2}$, with p prime ($C_n$ is the cyclic group of order $n$). I started setting (g,1) and (1,h) as generators of the ...
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122 views

Free abelian groups.

Is the following implication true? If this is the case, how can this be shown? If $G$ is a one-relator (neither power nor commutator relation), two-generator group, then $G/G'$ (where $G'$ is ...
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163 views

Find all numbers $n$ such that $S_7$ contains an element of order $n.$

Find all numbers $n$ such that $S_7$ contains an element of order $n.$ Identity is the only element of order $1.$So $n=1$ is possible. Case 1: Elements that can be written as a unique cycle of ...
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68 views

Is this group isomorphic to $\mathbb{Z}_k$? The integers with multiples of $k$ subtracted until you're under $r$.

Where as the elements of the ring of integers modulo $n$ can be found by taking the integers and subtracting multiples of $n$ until you're just under $n$, consider the structure formed by taking ...
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37 views

Proving a property of a finite subgroup

Prove that if $X$ and $Y$ are finite subgroups of a group $A$, then $|XY| = \frac{|X||Y|}{|X \cap Y|}$. I am not really sure how to even start this problem. It looks like something that would be not ...
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101 views

Group presentations and subgroups

How to prove that $G=\langle a,b,c\mid a^2 = b^2 = c^3 = 1, ab = ba, cac^{-1} = b, cbc^{-1} =ab\rangle$ has no subgroup of order $6$ without finding $G$? $\bf Edit$: Given that $|G|=12$.
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165 views

Find all homomorphisms in these three examples

How I should find all homomorphisms $$ f : G \rightarrow H $$ for examples: $$ i) \ \ G = (Z, +), \ \ H = (Q, +) $$ $$ ii) \ \ G = (Z_{15}, +_{15}), \ \ H = (Z_4, +_4) $$ $$ iii) \ \ G = Z_2 \times \ ...
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180 views

Abelian Groups and Number Theory

What is the connection between "Finite Abelian Groups" and "Chinese Remainder Theorem"? (I have not seen the "abstract theory" behind Chinese Remainder Theorem and also its proof. On the other ...
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35 views

Structure of stabilizer of nonsingular line in $\Omega(7,q)$

Let $q$ be a prime power, $\varepsilon=\pm$, and $M$ be the stabilizer of a nonsingular line in $\Omega(7,q)$ such that $M=\Omega^\varepsilon(6,q).2$. Then can we know more explicit structure of $M$? ...
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110 views

$G$-set terminology

When a group action $G \times X \rightarrow X$ is defined with a group $G$ and a set $X$, why is there not a special name for the set $X$? I know that this is referred to as a $G$-set, but the set $X$ ...
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126 views

Order of elements in $S_4$

Let $r(n) = \left| \left\{ \sigma \in S_4 : \mbox{ord} ( \sigma) = n \right\} \right|$. Is it true that: $r(2)>r(4)$ $r(4) > r(3)$ $r(1)+r(3) = r(2)$ $r(5) = r(6)$ I can write all elements ...
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84 views

Why are regular $p$-groups called “regular?”

In the concept of regular $p$-Groups, what does "regularity" stand for? What is "regular" in such groups? I would like to know idea behind defining these groups, and naming these groups "regular." ...
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224 views

multi-dimensional numbers

Questions: I am trying to derive a multi-dimensional number system. I know it is not the traditional way of doing things; my questions are: Is this valid? If not where are my mistakes? Has someone ...
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59 views

Find the inner and outermorphisms of a particular dihedral group

Given that |Inn($D_8$)| = 8 and |Out($D_8$)| = 2 where Out($D_8$) = Aut($D_8$)/Inn($D_8$) and $D_8$ = {e,r,$r^2$,..,$r^7$,s,sr,...,$sr^7$} we want to find Inn($D_8$) and Out($D_8$). We know that ...
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49 views

Showing a particular map is equivariant with respect to certain group actions

Let $A$ = {triangles in $\mathbb{R^2}$}. We can let $(x_1,y_1)$,$(x_2,y_2)$,$(x_3,y_3)$ be the vertices of the triangle. The group $GL(2,\mathbb{R})$ acts on $A$ by acting on the vectors of the ...
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435 views

Homomorphism between symmetric group and general linear group of order n. [closed]

I am having trouble proving the following: Show that $f: S_n \to GL_n(\mathbb{R}),\;\: f(x)=A_x$ is a homomorphism where $A_x$ is the permutation matrix associated with $x$. $S_n$ is the ...
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169 views

Direct product of cyclic group with itself

Let $R$ be the direct product of $C_p$ with itself. Show that $R$ is an abelian group of order $p^2$ and $R$ is not cylic.
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59 views

Prove normal of direct product of GL

Let $G=(\mathrm{GL}(2,\mathbb{R}) \oplus \mathrm{GL}(2,\mathbb{R}))$ and let $H = \{(A,B) \in G \mid\det(A)=\det(B)\}$. Prove that $H$ is normal in $G$. Mostly confused on what $G$ is.
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81 views

$ρ_ωG$ is a subgroup of $ρG$ containing $G$.

The $G_δ$-closure of $A$ in space $X$ is defined as the set of all points $x ∈ X$ such that every $G_δ$-subset of $X$ containing $x$ has a non-empty intersection with $A$. Now let $G$ be a ...
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305 views

Normal subgroup of direct product of two groups

The following is an exercise from Rotman's book: an introduction to the theory of groups. Let $N$ be a normal subgroup of $H\times K$. Show that either $N$ is abelian, or $N\cap H\neq 1$, or $N\cap ...
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110 views

Let $R$ be a ring with every element but $1$ having a left quasi-inverse. Then $R-\{1\}$ is a group under $a*b=a+b-ba$.

This question is related to exercise 1.51 from Rotman's "Introduction to the Theory of Groups". An element $a$ in a ring $R$ (with unit element $1$) has a left quasi-inverse if there exists an ...
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100 views

A character of an induced representation

I want a help to solve the following exercise from the book, Representation Theory, by Fulton and Harris. Exercise 3.19 (p.34) Let $H$ be a subgroup of a finite group $G$. Let $W$ be a representation ...
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94 views

A problem with the proof of a proposition

I have a problem with the proof of Proposition 5.1. of the article of Ito.(Noboru Itˆo. On finite groups with given conjugate types. I. Nagoya Math. J., 6:17–28, 1953.). I don't know what is "e" and ...
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60 views

Lemma to FToFinite Abelian Groups, clarification

~~ I am trying to understand this Lemma 8.4's proof. I understand what is happening until just after the line that says $$a=1a=(mu+nv)a=mua+nva.$$ At this point I do see why $mua \in G(p_1)$ since ...
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75 views

Show that the stabilizer is a prime subgroup

We define a subgroup $H$ as being convex if $g\in H\implies h\in H$ for all $1\leq h\leq g$. A convex subgroup $P$ is prime if for any two convex subgroups $X,Y$: $X\cap Y \subseteq P$ implies that ...
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103 views

Standard Wreath Product and Sylow Subgroups

A Sylow $p$-subgroup of $S_{p^r}$ is isomorphic with the standard Wreath Product $W(p,r) = (\cdots(C_p \wr C_p) \wr \cdots) \wr C_P)$, the number of factors being $r$. I have a great doubt as to ...
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82 views

Construct group $G=Q\ltimes Z_{7}$

I read a example. i have problem. Let $Q=\langle x, y| x^{9}=y^{3}=1, x^{y}=x^{4}\rangle$ and $Z_{7}$ be a group of order $7$. I know $\Omega_{1}(Q):=\Omega$ is an abelian group of order $9$. Since ...
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594 views

Groups of order 2k have a normal subgroup of order k and odd permutations

Okay, well, I have to show that, if $G$ is a finite group, such that, $|G|= 2k$, where $k$ is odd, then, translation by its element of order 2 is an odd permutation and $G$ must have a normal subgroup ...
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61 views

What is the group $\langle U, * \rangle$ where $U$ is the set of roots of unity and * is normal multiplication?

I'm having trouble understanding my textbook in this regard. Everything else seems to make sense, such as the group $\langle \mathbb{Z}_{1000}, + \rangle$ to list an example. In the text, it is ...
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93 views

$p$-component of finite abelian groups

On Wikipedia it is stated that for finite abelian groups, p-component is used to mean the Sylow p-subgroup, so the abelian group is the product of its p-components for primes p. Since every group can ...
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1k views

How to find generator in a finite group?what is generator?

Suppose that a group $Z_p=${$1,2,3......(p-1)$} where p is a prime number. How to Determine the generator/generators of this group? what are the possible method of finding it?
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147 views

If H, N are normal subgroups of G, then do all the commutators lie in the intersection?

Okay, I know that this is elementary, but, ah, well. How do I show that if N and H are normal subgroups of a finite group G with coprime orders, then, $xyx^{-1}y^{-1} \in H\cap N$ for all $x \in H, y ...
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79 views

What is the nature of the homomorphism of the semidirect product of two groups $H$ and $K$?

Let $H$ and $K$ be group and $H\rtimes_{\phi} K$ is the semidirect product of those group by the homomorphism $\phi:K\rightarrow Aut(H)$ Now , i have a main question about this function $\phi$ Does ...
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70 views

Finitely generated group and quadratic isoperimetric functions.

It is well known that a finitely generated group with simply connected cones has isoperimetric functions. Papasoglu in the paper "On the Asymptotic cone of groups satisfying a quadratic ...
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85 views

Show $G$ and $H$ are isomorphic

Let $G$ and $H$ be finite abelian groups of the same order $2^n$. If for each integer $m$, $$\left|\left\{x\in G\mid x^{\large 2^m}=1\right\}\right|=\left|\left\{x\in H\mid x^{\large ...
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57 views

How to choose a proper binary operation in a semigroup?

I am interested in generating a finite commutative semigroup which is not a group. And by generating I mean choosing a number $n$ (number of elements in a semigroup) and then defining a $n \times n$ ...
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181 views

multiplication table for mod n, n prime

How do I prove that multiplication table for mod n, where n is a prime gives rise to a latin square if the row and column of 0 is omitted? I need to prove for a fixed i, i * k is distinct when k runs ...
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335 views

two problems on Sylow's theorem

How many elements of order $5$ does a non-cyclic group of order $55$ have? Let $G$ be a group of order $105$. Show that $G$ has a subgroup of order $35$. By Sylow's theorem it has two ...
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88 views

Getting the multiplication table in Magma

I have described a group in Magma by specifying a presentation: ...
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91 views

Is this graph coloring problem solved correctly?

On this Wikipedia page about Burnside's lemma, it is calculated that there are 57 rotationally distinct colorings of the faces of a cube with three colors. I'm confused by the way it is done. They ...
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108 views

Subdirect product of simple groups

Let $G$ be a subdirect product of simple groups $G_i$ for $i\in I$. I want to show that if $1\neq x\in G$, then there exists a maximal normal subgroup $N$ such that $x\notin N$. First of all, I know ...
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60 views

On the Frattini Subgroup

For a prime $p$, let $H=\{x\in \mathbb{C}\colon x^{p^n}=1 \mbox{ for some } n\geq 1\}$ be the Prufer $p$-group, $C_2=\langle y\colon y^2=1\rangle$, and $G=H\oplus C_2$. Then $H$ is the unique maximal ...
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94 views

Given a group homomorphism $f:G\to H$, if $m$ is relatively prime to $|H|$ and $x^m\in\ker f$, then $x\in \ker f$

Let $f:G\to H$ be a homomorphism, and let $m$ be an integer such that $m$ and $|H|$ are relatively prime. For any $x \in G$, if $x^m \in \ker f$, then $x \in \ker f$. My proof step: if $x^m \in ...
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150 views

Hall $\pi$-subgroup and $HN=G$

Let $\pi$ be any set of prime numbers. A finite group $H$ is a $\pi$-group if all primes that divide $|H|$ lie in $\pi$. If $|G|<\infty$, then a Hall $\pi$-subgroup of $G$ is a $\pi$-subgroup ...
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230 views

Show that a p-group has a faithful irreducible representation over $\mathbb{C}$ if it has a cyclic center

A p-group is a group of order $p^d$ where p is a prime. If the center has order $p^m$ (since its order must divide the order of the group) then we have a one dimensional faithful irreducible ...
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143 views

Determining the group homomorphism in semidirect product

We know that if $N$ is a normal subgroup, $H$ is a subgroup, and $\varphi$ is the group homomorphism such that $\varphi:H\to$Aut$(N)$. And this gives a unique group, called the outer semidirect ...