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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On the number of minimal normal subgroups w.r.t. to a maximal subgroup which is simple and non-abelian

Let $G$ be a finite group, and $L$ a maximal subgroup of $G$. If $L$ is non-abelian and simple, then in $G$ there exists at most two minimal normal subgroups. What I got: Suppose we have three ...
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130 views

How to show that the intersection of all neighbourhood of $0$ in a topological group is a subgroup?

Let $H$ be the intersection of all neighbourhood of $0$ in a topological group $G$. How to show that $H$ is a subgroup? I tried to use the continuity of multiplication and inverse. But not ...
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1answer
88 views

Graph Jacobian (Sandpile group) usages

Let $\Gamma$ be a graph (say, finite) and $S_\Gamma$ be it's Jacobian (also known as the sandpile group or Picard group). I'm wondering about what fundamental things one can learn about $\Gamma$ from ...
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46 views

When a multiplicative subgroup of a field generate a field?

Is it possible to find a field $F$ of prime characteristic which contains a non-trivial cyclic infinite subgroup $\langle x\rangle$ of $F^\times$ (the multiplicative group of $F$) such that the ...
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2answers
71 views

Equation over finite group with presentation $G=\langle a,b|R\rangle$

This question arised like a curiosity, I've been trying to find out information about the solution (after trying to solve it by myself) with no success. The question is: Given a finite group with ...
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327 views

Is $A_n$ non-abelian for $n= 3$?

In the book, it is asked to show that $A_n$ is non-abelian for $n ≥ 4$. Which may imply that it is abelian for $n=3$. Is that so? because $(13)(12)\ne (12)(13)$. Hence is it true to write: $A_n$ is ...
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1answer
44 views

Prove that $\langle S \rangle = G$

Let $S\subset G$ be a finite group such that $\#2S\gt\#G$ Prove that $\langle S \rangle = G$ My idea was to start by taking $a \in G$ and proving that $ax^{-1}$ is also in $G$ if $x$ is in $S$, but ...
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58 views

Order of Aut$(D_4)$

How can I prove that order of Aut$(D_4)$ is 8. Let we show $D_4$ as $\{e,\sigma,\sigma^2,\sigma^3,\tau,\tau\sigma,\tau\sigma^2,\tau\sigma^3\}$ and ...
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1answer
49 views

$G=HK$ then the index of a subgroup is determined by $H$ and $K$

Let $G=HK$ s.t. $H\cap K=1$ and let $R$ be a any subgroup of $G$. I wonder necessary and suffucient condition for the equality, $$|G:R|=|H:H\cap R||K:K\cap R|$$ Note that if $H$ and $K$ are normal ...
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51 views

Full subcategories of $\mathsf{Grp}$ with epis/monos which are not surjective/injective

I read that there exist full subcategories of $\mathsf{Grp}$ in which there are epis which are not surjective, and ones in which there are monos that are not injective. I'm confused because they're ...
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110 views

Show that there is exactly one binary operation making the set $\{e, x, y\}$ a group with $e$ the identity element

Let $S=\{e, x, y\}$ be a set of three elements. Show that there is exactly one binary operation making the set $S$ a group such that $e$ is the identity element. So I have produced the ...
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1answer
46 views

Are Sylow p-subgroups in conjugate?

Let $G$ be an infinite group and $p$ be a prime number. Let $\mathscr{C}$ be a chain of p-subgroups of $G$ ordered by inclusion. Then, for every element of the union of $\mathscr{C}$ has an order ...
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1answer
41 views

Eigenvalues of operator on $S_n$'s group algebra

Take the group algebra of the symmetric group $S_n$ (or equivalently consider $S_n$'s regular representation) - I guess over $\mathbb{C}$. If $e_{i,j} \in S_n$ denotes the element which swaps only ...
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97 views

Inequivalent representations of a finite group

I'm looking for this result: A finite group has only finitely many inequivalent representations of given degree over a field of characteristic $0$. Do someone know where I can find a proof of ...
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1answer
109 views

Classifying groups of order $2p^2$

I want to solve the following exercise from Dummit & Foote's Abstract Algebra text (p. 185 Exercise 15): Let $p$ be an odd prime. Prove that every element of order $2$ in $GL_2(\mathbb{F}_p)$ ...
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206 views

Finding all the group elements of a certain order of a finite group

Consider a group $G=\langle P, Z, Q \rangle$ generated $P,Z,Q$ where $$Z=\left[ {\begin{array}{ccc} -1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & -1\\ \end{array} } \right] ...
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84 views

Fiber product of non-abelian groups.

I am trying to understand whether surjectivity is needed for a fiber product of non-abelian groups to exist. I seem to have checked that the usual construction works for groups without any ...
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1answer
34 views

If G is a compact semisimple Lie group and Z is its center, is G/Z always compact?

The title pretty much sums up the question: Suppose $G$ is a compact semisimple Lie group with center $Z$, The question is if $G/Z$ is always compact? or, under which conditions will it be compact?
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1answer
227 views

class equation of order $10$

Is it a class equation of order $10$ $10=1+1+1+2+5$. As far as I know for being a class equation each member on RHS has to divide $10$ and should have at least one $1$ on RHS, which is ...
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72 views

Why is every coset in G a subset of G?

Suppose $G$ is a group and $H$ is a subgroup of $G$. $Ha$ is a right coset of $H$ in $G$. According to the Dover Book of Abstract Algebra p. 127, "Every coset in $G$ is a subset of $G$." I understand ...
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250 views

Online Archive of Master Thesis

I am thinking about taking the thesis route to complete my master in pure math. In anticipation of these in the coming semesters, here are my questions: (1) Do you know of any links to archive of ...
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55 views

Inequalities in groups.

I learnt that $(\mathbb{R},\times) < (\mathbb{C},\times)$, Which means the first is a subgroup of the second one. But in the first group inequality is defined, while it's not in the latter. This ...
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73 views

Proof Using Lagrange's Theorem

I am working on a problem in Kurzweil & Stellmacher's introductory finite group theory that looks like this: Let $A, B$, and $C$ be subgroups of the finite group $G$. Prove that if $B \leq A$, ...
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111 views

Proving the existence of a bijection between $U_{mn}$ and $U_m \times U_n$ where $(m,n)=1$ , there by proving Euler $\phi$ is multiplicative

Without proving before hand that Euler's phi $(\phi)$ function is multiplicative , can we prove that there is a bijection between $U_{mn}$ and $U_m \times U_n$ , for any pair of relatively prime ...
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80 views

Set of conjugates of element cannot be contained in the union of two proper subgroups

Edit: Using the excellent hints provided by @SMM, I was able to solve the problem. See my answer below. I've been thinking off and on about this problem over the last couple of days and would ...
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1answer
88 views

Proving that a subgroup $|H|=p^k$ is a Sylow subgroup of $|G|=p^km$, $m\nmid p$

I'm attempting to prove Sylow's theorems following the sketch described in the Wikipedia article, but I've run into a little hitch since the theorems are presented in a few slightly different forms in ...
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1answer
33 views

Why it is central in $\mathbb {Z}[G]$?

In proposition 4.17, why is $P$ an central element?
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31 views

Extension of group homomorphsims [duplicate]

Let $G$, $G'$ be groups, and $H$ a subgroup of $G$. Given a homomorphism $f:H\longrightarrow G'$, does there exist a homomorphism $\tilde{f}:G\longrightarrow G'$ such that $\tilde{f}|_{H}=f$?
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39 views

When does $S_n$ have exactly one subgroup of index $2$?

When $n\geq 5$ we can assume there is another one besides $A_n$, they are both normal, so the intersection is normal in $S_n$ and also in $A_n$, contradicting that $A_n$ is simple. For $n=1$ it's ...
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78 views

Isomorphic Coxeter Groups

Is it true that two Coxeter groups having the same Coxeter matrix (equivalently, the same Coxeter graph) isomorphic? Because otherwise, the definition of a Coxeter group from its Coxeter matrix does ...
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1answer
57 views

Is the solution of this trivial group theory exercise correct?

If $G$ is a finite group, prove that, given $ a \in G $, there is a positive integer $n$, depending on $a$, such that $a^n=\bf1$, where $\bf1$ is the identity element of the group. This is my proof: ...
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2answers
85 views

Problems about generators for Sylow p-subgroups

There are several problems I met asking to find the generators for some different Sylow $p$-subgroup. $(i)$ a Sylow 2-subgroup in $S_{8}$; $(ii)$ a Sylow 3-subgroup in $S_{9}$; $(iii)$ a Sylow ...
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1answer
87 views

Subgroups and an union of orbits

I have to prove or disprove the following statement: If a group $G$ acts on a set $X$, then every subgroup $H$ of $G$ acts on the set $X$ as well, and every orbit of the action $G$ on $X$ is an ...
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2answers
51 views

Group theory, the squares of G

We have a group $G$ with a subgroup $G_2$, which is defined by $G_2:=\{g^2|g \in G \}$. I have to prove that i) $G_2\triangleleft G$ ii) all elements of $G/G_2$ have order $\leq2$ iii)if ...
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1answer
76 views

Finding a surjective homomorphism

I have to show that for all groups with $2007(=3^2\times223$) elements that there exists a surjective homomorphism to a group of 9 elements. Obviously a group with 2007 elements has a subgroup of ...
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1answer
101 views

A question in character theory of finite groups

This is not a homework. I am a beginner in studying the representation theory and character theory and I am doing some exercises from "A Course in the Theory of Groups by Robinson" and "character ...
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1answer
151 views

Centralizer, Normalizer and Conjugate

I am looking at Group Theory notes on Centralizer and Normalizer for next semester and come up with this question: Let $H$ be a subgroup of $G$, and let $g$ be an element in $G$. Show that $$(a)\ ...
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1answer
140 views

Generalized Cauchy's theorem (group theory)?

I think there is a corollary for Cauchy's theorem such that a finite group $G$ contains a subgroup of order $n$ for each divisor $n$ of $|G|$ with a certain condition (I can't remember what it was). ...
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1answer
75 views

Show that there are at least $2$ elements in $U(n)$ such that $x^2=1$.

I am working on some exercises in Joseph Gallian's Contemporary Abstract Algebra. I came upon the following: Show that there are at least $2$ elements in $U(n)$ such that $x^2=1$, for $n>2$ ...
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1answer
44 views

for $n \ge 3$, $S_n$ is isomorphic to its group of inner automorphisms

How would I go about showing that for $n \ge 3$, $S_n$ is isomorphic to its group of inner automorphisms?
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1answer
164 views

Select k no.s from 1 to N with replacement to have a set with at least one co-prime pair

Given $1$ to $N$ numbers. You have to make array of $k$ no.s using those no.s, where repetition of same no. is also allowed, such that at least one pair in that chosen array is co-prime. Find no. of ...
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A group in between the commutator subgroup and the original group must be normal

Let $C$ be the commutator subgroup of a group $G$. then by some easy arguments, we know that $1$. $C$ is normal in $G$ $2$. $G/C$ is abelian $3$. If $N$ is normal in $G$ and $G/N$ is abelian, then ...
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56 views

No character theory, any representation ${\bf{GL}}_N(\mathbb{C})$ is reducible, upper bound

Let $G$ be any finite group. How do I show without character theory that there is a number $N = N(G)$ so that any representation $\rho: G \to {\bf{GL}}_N(\mathbb{C})$ is reducible (and finding an ...
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2answers
24 views

Product of two arbitrary elements in $O(2)$

An exercise in a book I'm reading asks to describe the product of two arbitrary elements in $O(2)$. I would like to solve the exercise but I got stuck. I know that an element in $SO(2)$ can be written ...
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2answers
94 views

Isomorphism type of a finite group with respect to multiplication modulo 65

I'm the same guy revising for my group theory exam and posted a few days ago. I'm at the chapter on Finitely Generated Abelian Groups, and my prof gave this example which I don't quite understand: ...
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1answer
47 views

For which $n\in\mathbb{Z}^+$ is there a $\sigma\in S_{14}$ such that $|\sigma|=n$?

For which $n\in\mathbb{Z}^+$ is there a $\sigma\in S_{14}$ (the group of permutations on $\{1,2,\dots,14\}$) such that $|\sigma|=n$ (where $|\sigma|$ is the order of $\sigma$)? I know you could just ...
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1answer
50 views

Calculate the number of conjugacy classes of $G$ with $|G| = p^4$ with $|Z(G)|=p^2$

Let $p$ be a prime and $G$ be a group of order $p^4$ such that $|Z(G)|=p^2$. Calculate the number of conjugacy classes of $G$. I couldn't think of much except for, if $G$ acts on itself by ...
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44 views

Normal subgroup of 2-transitive group

I want to show that if $G$ acts faithfully and $2$-transitively on a set $X$, and $N$ is a non-trivial normal subgroup of $G$, then $N$ acts transitively on $X$. I'm convinced that I have to use the ...
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2answers
105 views

$SU(2)$ acting by conjugation, decomposition into irreducibles

I am attempting past exam questions of the Cambridge Math Tripos. I know how to do the first few parts, which involves giving the irreducible representations of $U(1)$ and $SU(2)$. But I am not sure ...
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1answer
64 views

What is the Cayley graph of $(\Bbb Z/2\Bbb Z)\times(\Bbb Z/2\Bbb Z)$?

I get that the presentation of the new group, with respect to two generators, would be $(x,y \;|\; x^2= y^2=1)$ but I'm not sure how the actual graph would look. Would it consist of an infinite ...