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

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The Schur Theorem

The Schur Theorem: if $\left\vert G:Z\left( G\right) \right\vert $ is finite, then $G^{\prime }$ is finite. My question is: if $1\not=N\trianglelefteq G$ such that $\left\vert G:NZ\left( G\right) ...
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147 views

Original article on the Grothendieck group

Is there someone who knows the title of the original publication of Grothendieck on the construction of the Grothendieck group? Thanks in advance.
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0answers
65 views

Pi-subgroup part-2

Let $\mathbb{P}$ is a set primes numbers, $\pi \subseteq P$ and $\pi ^{\prime }=P-\pi $ Let $G$ abelian and $O_{\pi }\left( G\right) =\left\langle N~;~N\trianglelefteq G\text{ and }% N\text{ is }\pi ...
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1answer
245 views

Abelian subgroup of a group of order $2002$

Another unsolved question from my studying for quals - Show that if $G$ is a group of order $2002=2\cdot 7 \cdot 11 \cdot 13$, then $G$ has an abelian subgroup of index 2. I know it has to do with ...
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1answer
53 views

Pi-subgroup in group

Let $\mathbb{P}$ is a set primes numbers, $\pi \subseteq P$ and $\pi ^{\prime }=P-\pi $ Let $O_{\pi }\left( G\right) =\left\langle N~;~N\trianglelefteq G\text{ and }% N\text{ is }\pi ...
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2answers
3k views

Examples on isomorphism and homomorphism

Could someone please explain to me how isomorphisms and homomorphisms work? For an isomorphism, I know we need to follow the following four steps: $\ \ \ $1) define a candidate, $\ \ \ $2) show ...
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3answers
707 views

Can a infinite group be a finite union of proper subgroups?

We know that a group can not be written as a union of two proper subgroups and obiously a finite group can be written as a finite union of proper subgroups.So I want to ask if a infinite group be ...
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7answers
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Is addition more fundamental than subtraction?

When I followed an introductory! course group theory and throughout all my Math courses as a physicist, subtraction was always defined in terms of the inverse element and addition. Is this the only ...
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1answer
752 views

Prove that $ U(n^2−1)$ is not cyclic

Prove that $U(n^2−1)$ is not cyclic, where $U(m)$ is the multiplicative group of units of the integers modulo $m$.
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966 views

Subgroup of $\mathbb{Q}$ with finite index

Consider the group $\mathbb{Q}$ under addition of rational numbers. If $H$ is a subgroup of $\mathbb{Q}$ with finite index, then $H = \mathbb{Q}$. I just saw this on our exam earlier and was stumped ...
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1answer
228 views

When is $\mathbb{Z}$ a flat $\mathbb{Z}G$-module?

Suppose that $\mathbb{Z}$ is a flat $\mathbb{Z}G$-module for a group $G$. Question: Is $G$ the trivial group ? Nb. I know that the question can be answered affirmatively if $G$ is finitely ...
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1answer
132 views

Congruence inequality

Given $n>2$, by calculation or otherwise deduce that $5^{2^{n-3}} \neq -1 \pmod {2^n}$ Note:The problem arose when I tried to deduce $\langle5\rangle \cap \langle2^n-1\rangle=\{1\}$ in the group ...
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3answers
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Visualizing quotient groups: $\mathbb{R/Q}$

I was wondering about this. I know it is possible to visualize the quotient group $\mathbb{R}/\mathbb{Z}$ as a circle, and if you consider these as "topological groups", then this group (not ...
2
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1answer
769 views

One step subgroup test help [duplicate]

Possible Duplicate: Basic Subgroup Conditions could someone please explain how the one step subgroup test works, I know its important and everything but I do not know how to apply it as ...
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2answers
889 views

Sylow 7-subgroups in a group of order 168

Another question from my qual studying that's been stumping me. I'm still a little lost on normalizers. The question is: Let G be a group of order 168, and let P be a Sylow 7-subgroup of G. Show ...
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1answer
218 views

Conjugate elements of G that are in the center of a Sylow p-subgroup

I've had a lot of trouble with this question, as well as my advisor. Let x and y be two elements of $Z(P)$, where $P$ is a Sylow $p$-subgroup of $G$. If $x$ and $y$ are conjugate in $G$, prove that ...
3
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2answers
173 views

An element conjugate to an odd number of elements

I have the following problem I've encountered while studying for quals that I just can't seem to tackle: (revised, apologies - left out an important detail) An element g of order greater than 2 such ...
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1answer
127 views

Unipotent action of pro-$p$-group

Say $p$ and $\ell$ are distinct prime numbers. Let $G$ be a pro-$p$-group which acts continuously on a finite-dimensional $\mathbb{Q}_\ell$-vector space $V$. Assume that the action of $G$ on $V$ is ...
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2answers
210 views

Where is the symmetric group hidden in the Yoneda lemma?

In extension to the question Yoneda-Lemma as generalization of Cayley`s theorem?, can someone point out to me where, in the categorical notation and analyzation of the Cayley's theorem, the symmetrc ...
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2answers
1k views

$gHg^{-1}\subset H$ whenever $Ha\not = Hb$ implies $aH\not =bH$

If $H$ be a subgroup of a group $G$ such that $Ha \not=Hb$ implies that $aH\not=bH$.Then how can I show that $gHg^{-1}\subset H$ $\forall$ $g\in G$? I do not think I have made any progress.However, ...
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589 views

Every Ring is Isomorphic to a Subring of an Endomorphism Ring of an Abelian Group

Show that for every ring $(R,+,\cdot)$, there is an abelian group, $(A,+)$, such that $R$ is isomorphic to a subring of $(\operatorname{End}(A),+,\circ)$. $(\operatorname{End}(A),+,\circ)$ is the set ...
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1answer
227 views

Complexity of finite group isomorphism problem

Consider the next decision problem: Given two finite groups represented by their multiplicity table, determine if they are isomorphic or not. Clearly, this problem belongs to NP since given a witness ...
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1answer
760 views

Some questions about representations of $SO(6)$

I would like to know the proof/explanation for the following three properties of the representation of $SO(6)$, What is the importance of symmetric traceless tensors of arbitrary rank w.r.t $SO(6)$ ...
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0answers
109 views

Generating set withous elements of order 2

I proved that every group acts freely by left translation on its Cayley graph if and only if the generating set $S$ of the group $G$ has no elements of order $2$. Is it true that every finitely ...
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0answers
95 views

In finite groups does counting orders of elements is enough to determine if they are isomorphic [duplicate]

Possible Duplicate: Three finite groups with the same numbers of elements of each order Suppose that we have two finite groups $G$ and $H$ such that for each $n\in\mathbb{N}$ ...
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1answer
113 views

Orbits of $\mathbb{Z}/n\mathbb{Z}$ under automorphism subgroup action

Let $A\leq \rm{Aut}(\mathbb{Z}/n\mathbb{Z})$. I am interested in the orbits of $\mathbb{Z}/n\mathbb{Z}$ under the action of $A$, i.e. the sets $$\{\sigma i : \sigma \in A\},$$ where $i\in ...
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2answers
339 views

Order of a set $X$ acted upon transitively by the Symmetric Group

Suppose the symmetric group $S_n$ acts transitively on a set $X$, i.e. for every $x, y \in X$, $\exists g \in S_n$ such that $gx = y$. Show that either $|X| \le 2$ or $|X| \ge n$. Small steps ...
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2answers
2k views

What kind of work do modern day algebraists do?

Often times in my studies I get the impression that algebra is just a tool to help with other branches of mathematics, like algebraic geometry, algebraic number theory, algebraic topology, etc. How ...
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1answer
184 views

Uniqueness of conjugates of a subgroup.

This question is partly influenced by the question: Are two subgroups that contain a common element conjugate iff they are conjugate under the normalizer? If we have an arbitrary finite group $G$ ...
3
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1answer
219 views

Is an exterior algebra a skew group ring?

Can an exterior algebra $$ k\langle x_{1},\dots,x_{n} \rangle/(x_{1}x_{2}-x_{2}x_{1},\dots,x_{1}^{2},\dots) $$ can be seen as a skew group algebra? A skew group ring is defined for example in the ...
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Smallest generator of monoid of all groups under $\times$

What is the smallest set of groups $S$, such that for any group $G$ there exists either $H \in S$ or $H = H_1 \times H_2 \times \dots \times H_n$ for $H_i \in S$ such that $G$ and $H$ are isomorphic. ...
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2answers
550 views

Basic group theory exercise

I've been reading some stuff about algebra in my free time, and I think I understand most of the stuff but I'm having trouble with the exercises. Specifically, the following: Prove that a nonempty ...
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1answer
524 views

Fundamental and the anti-fundamental representation of $U(n)$

I guess that conventionally one thinks of the fundamental representation and the anti-fundamental representation of $U(n)$ as the complex $n-$dimensional representation and its complex conjugate. ...
4
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1answer
239 views

Groups quasi-isometric to $\mathbb{Z}^n$

I am interested by the following result: A groups quasi-isometric to $\mathbb{Z}^n$ is virtually $\mathbb{Z}^n$. I know the article Harmonic analysis, cohomology, and the large-scale geometry of ...
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1answer
158 views

Understanding an Outer Automorphism of $S_6$

In an article (paper), there is a description of an outer automorphism of $S_6$. There are six pentagons, arranged with a rule, with vertices $1,2,3,4,5$. Any permutation of these vertices will ...
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1answer
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What is the algebraic structure of functions with fixed points?

So I just noticed that the set of functions with a fixed point $$f(x_0)=x_0,$$ are closed under composition $$(f\circ g)(x):=g(f(x)),$$ and with $e(x)=x$, the inverible functions even seem to form ...
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2answers
323 views

How many small cancellation groups are there?

It is known that there are uncountably many groups with two generators. But what about the restriction to small cancellation groups? Are there countably or uncountably many small cancellation groups? ...
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1answer
232 views

Permutation Groups

I recently saw a question in a Text book, which asks to prove that "The group of symmetries of the polynomial $x_1x_2 + x_3x_4 + x_5x_6$ is a subgroup of $S_6$ of order $48$". (By the group of ...
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1answer
96 views

Latin squares of even order with all cells only participating in one subsquare.

For even ordered Latin squares, we can create squares in which every cell participates in a $2\times 2$ subsquare by using a simple circulant as for $n=6$ below: ...
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2answers
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Product of right cosets equals right coset implies normality of subgroup

I cannot see how to find a way to prove that if $H$ is a subgroup of $G$ such that the product of two right cosets of $H$ is also a right coset of $H,$ then $H$ is normal in $G.$ (This is from ...
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94 views

Notations for groups of order $p^3$

Are there any relatively common notations for the two non-isomorphic nonabelian groups of order $p^3$ where $p$ is a prime number? I remember reading some notations like $p_+^{1+2}$ and $p_-^{1+2}$. ...
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1answer
74 views

Showing elements with certain properties are in a normal subgroup

I'm preparing for an exam, and this little guy just had me stumped: Let $G$ be a group, $N\subset G$ a normal subgroup of $G$, $x,y,z \in G$, and $x^3 \in N$, $y^5 \in N$, $zxz^{-1}y^{-1} \in N$. ...
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Kernel of Group Action

this is my first post here. I have a question regarding a proof in Algebra by Hungerford: Let $G$ be a group and $H$ a subgroup of $G$. Let $S$ be the set of all cosets of $H$, where $G$ acts on ...
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1answer
140 views

Morphisms between cyclic groups

I'm trying to solve a group theory question involving morphisms: How many different morphisms do there exist from $ C_n $ to $ C_m $? Am I correct in saying if $f$ is a morphism, then $f(0) = 0$ ...
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1answer
314 views

How to find all the subgroups of a group that contain a given subset?

Given a group G of order n and a subset S of G such that |S|=m. What is the best algorithm for generating all the subgroups of G that contain S? How the complextity of such algorithm depends on n and ...
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1answer
346 views

application of group isomorphism theorem?

Let $M$ and $N$ be normal subgroups of $G$. Find a homomorphism $f:G \rightarrow \frac{G}{M} \times \frac{G}{N}$ and use this to prove that $\frac{G}{M \cap N}$ is isomorphic to a subgroup of ...
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1answer
106 views

A basic group question

Let $G=\{0, \cdot\}$. I'm arguing with someone over if $G$ is a group with the regular multiplication since I don't see why it isn't. Addition: Now, $G=\{\mathbb{Z},\triangle \}$ with $x \triangle ...
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2answers
228 views

Generators of a cyclic group

In a paper there is a lemma: Let $G= \langle a,b \rangle$ be a finite cyclic group. Then $G=\langle ab^n \rangle$ for some integer $n$. The proof is omitted because it's "straightforward" but ...
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is a group that eliminates quantifiers in the group language Abelian?

Let $G$ be a group. Consider it as a structure in the language of groups $L=(e,\cdot, (-)^{-1})$. Suppose that $G$ eliminates quantifiers in this language. Is it true that $G$ must be Abelian then? ...
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176 views

Properties of homomorphisms of the additive group of rationals

Let $f : (\mathbb{Q},+) \longrightarrow (\mathbb{Q},+)$ be a non-zero homomorphism. Can we conclude that $f$ is bijective (or, if that fails, that $f$ is injective or surjective)? Context The ...