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

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Question About Group Theory Notation

I am having trouble understanding what "Universal Cover of $\mathbb{Z} \times \mathbb{Z}$" mean exactly. Thanks
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Why is every simple object in the category of abelian groups simple in the category of groups?

I'm not asking for a proof that every simple abelian group is simple; that's a fairly trivial question. To lead into my real question, I started thinking about this question: Let $\cal C$ be a ...
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On the meaning of “Class of finite groups”.

What do we mean precisely when we speak about a class of finite groups? Is this simply a collection of some finite groups, maybe collected with a criterion (example: the class of all finite cyclic ...
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Wreath product (Rotman, p.173)

I have a question about the explanation of wreath product in Rotman's "An introduction to the theory of groups (4th ed.)." Let $D$ and $Q$ be groups, $\Lambda$ be a $D$-set, $\Omega$ be a finite ...
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Stirling numbers and Power Group Enumeration

The following question is a reference request concerning a derivation of the EGF for the Stirling numbers of the second kind by Power Group Enumeration / Burnside's Lemma, which is $$\sum_{n\ge 0} ...
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continuation of problem subgroup isomorphic

This subject is associated with Does group $A_6$ contain subgroup isomorphic with $S_4$ Hint: Take $S_4$ on the first four elements of the set permuted by $A_6$. Some of these are odd ...
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Automorphism Tower

Let $G$ be a group. Now consider $\operatorname{Aut} G$ then $\operatorname{Aut} (\operatorname{Aut} G) = \operatorname{Aut}(2G)$, then $\operatorname{Aut}(\operatorname{Aut}(\operatorname{Aut} ...
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find order of function -

Let $f(n) = x$, where $x^x = n$. Find order of function $f$. And it is strange for me. Why ? Firslty, I don't see any group. Ok, but look at my reasoning. What is definition of order ? $ord(a) = ...
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Abstract Algebra 1 about permutation

I have a question from exam and I really don't know how to solve it, and tried a lot. Say I have a permutation $a=(1 6)(2 5 7)(3 8 4 9)$ so that $a \in S_9$ and I want to find how many permutations ...
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Can the integral group ring construction be extended to groupoids in such a way that it provides a functor from Groupoids to Rings?

If $G$ is a group, and $R$ is a ring, one can construct the group ring $R[G]$, which is a free $R$-module with basis $G$, and with multiplication coming from the original multiplication of $G$. ...
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How to prove that $\text{Aut}(\mathbb{Z}/p \times \cdots \times \mathbb{Z}/p )\simeq GL_n(\mathbb{Z}/p)$

Anyone can give me any clue about the proof of $\text{Aut}(\mathbb{Z}/p \times\cdots \times \mathbb{Z}/p )\simeq GL_n(\mathbb{Z}/p)$? Thank you.
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Why Lagrange's Theorem implies $|G/K|$ divides $p!$ (proof of Corollary $5$ on page $120$ of Dummit and Foote)

In the proof of Corollary $5$ on page $120$ of Dummit and Foote's Abstract Algebra, I don't understand why Lagrange's Theorem implies that $|G/K|$ divides $p!$. Here is an excerpt.
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coloring cube, additional constraint, three colors

I have to paint nodes of cube such that opposing nodes has the same color. We consider identical cubes such that rotatating. My result is $15$ Is it correct ? Ok, I 'll add my way to get a result. ...
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Determining elements of group from generating relations

Is there an algorithm for determining all the elements fo a finite group from its generating relations? For example, let group $G$ have the generating relations $p^3 = q^2 = (qp)^2 = 1$. I see that ...
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Determine if the following sets with the given operations are groups.

For the first three I just want to confirm if my answers are correct. Hence I will state if it is or isn't a group. If it isn't a group I will state what property it fails. The problem I do seek ...
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lattice symmetries portrayed in animation

Below is an animated gif by Dave Whyte: What is the orbifold/fibrifold notation for the symmetries of the lattice depicted below? (a la /The Symmetries of Things/? Are there related theta function ...
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Prove A(S1) is isomorphic to A(S2)

I got a problem in assignment, although I have done this but just want to know if proof is correct. ...
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Chernikov Groups and layer-finite groups

A group $G$ is Chernikov if it has a subgroup of finite index that is a direct product of finitely many groups of $C_{p^\infty}$ for various primes $p$ (quasicyclic $p$-groups). The subgroup of finite ...
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Does group $A_6$ contain subgroup isomorphic with $S_4$

Does group $A_6$ contain subgroup isomorphic with $S_4$ ? The only thing that I ask for is any clue.
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A Group Having a Cyclic Sylow 2-Subgroup Has a Normal Subgroup.

I want to prove the following: Let $G$ be a group of order $2^nm$, where $m$ is odd, having a cyclic Sylow $2$-subgroup. Then $G$ has a normal subgroup of order $m$. ATTEMPT: We ...
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Group's morphisms

I know that the constant map equal to $1$ and the signature are two group's morphisms from the group of permutations $(\mathcal S_n,\circ)$ to the group $(\Bbb C^*,\times)$. My question is: Can we ...
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Proving a group, $G$, is a group action onto some set, $X$

I want to prove that a function defines a group action: We have group $G$ of diagonal $2\times 2$ matrices under matrix multiplication, and the set $X$ of points of the Cartesian plane, eg: ...
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Generalization of a property of proper conjugate subgroups of a finite nilpotent group

Let $G$ be a finite nilpotent group and $H, K$ be two proper non-maximal subgroups of $G$ such that $H\not\leq\Phi(G), K\not\leq\Phi(G)$ and $\langle H, K\rangle < G$. Does there exist a maximal ...
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Number of actions of $\mathbb Z$

Let $X$ be a finite set. Determine the number of actions of $\mathbb Z$ on $X$. If $X$ is a finite set with $|X|=m$, then $|\{f:X \to X : \text{f is bijective}\}|=m!$. Finding the number of actions ...
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Normal subgroup acting on a set

I am trying to solve the following problem: Let $G$ be a group acting on a set $X$ and let $S \lhd G$. Determine the necessary and sufficient conditions so that there exists an action of $G/S$ on $X$ ...
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Existence of a finite group union of self normalizing subgroups

Does a finite group G union of self normalizing subgroups such that the intersection of any two of these subgroups is equal to the unit of group G exist? I don't think so, but I can't prove it. Thank ...
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group of permutation - cycles of length $n$ - order of generated group

We have all cycles of length $n$. For example when $n=4$ our set is following form: $$G = \{(1234);(1243);(1342);(1324);(1432);(1423)\} $$ Our task is to find order of the group generated by the set ...
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Order of a group $G$

A finite group contains an element $x$ of order $10$ and also an element $y$ of order $6$. What can be said about the order of $G$ ? By Langrange's theorem, we establish a corollary stating that the ...
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Exercise on Cyclic Group

Let $G$ be a finite cyclic group of order $n$. Let $a$ be a generator. Let $r$ be an integer $\neq 0$, and relatively prime to n. Show that $a^r$ is also a generator of $G$. Proof: $G=\langle ...
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Conjugation (Group vs. Algebraic)

I am just starting to learn about groups, and the concept of conjugation came up. I was wondering what the relationship was, if any, between conjugation in the group sense and conjugation in the ...
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About $ AGL(1,16)$

I wanted to know orders of all subgroups of the group $AGL(1,16)$ (of order 240). Indeed, regarding to the problem mentioned in ...
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Showing that $\mathbb{H}^{*}$ maps onto $\mathrm{Aut}(\mathbb{H})$

To show that $\mathbb{H}$ maps onto $\mathrm{Aut}(\mathbb{H})$, where $\mathbb{H}$ is the Hamilton quaternions, I thought it'd be pertinent to show first that the subgroup of inner automorphisms of ...
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Is $SL_2(\mathbf{Z}/n\mathbf{Z})$ generated by the elementary matrices?

By the elementary matrices I mean the matrices with diagonal elements $1$ and at most one nonzero element off-diagonal. I have seen it claimed that this is true, but no proof was given. I know that ...
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Quote objects in concrete categories

I refer to 'Analogy of ideals with Normal subgroups in groups' which was a very enlightening question for me. When I was young I was too avid on abstract algebra and I did too many courses at the same ...
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map between classifying spaces induced by group homomorphism

Let $\Sigma_n$ be the permutation group of order $n$. Then the regular representation of $\Sigma_n$ gives an injective homomorphism $f:\Sigma_n\to O(n)$. Why $f$ induces a map between their ...
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How to show that any non-trivial subgroup of G is cyclic

My assignment Question is: Let G be a group of order pq where p and q are primes. Show that any non-trivial subgroup G is cyclic. Now according to Cosets and Lagrange theorem, is state that the ...
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Is it possible that a left coset of $H$ contains more than one right coset of $H$?

Let $H$ be a subgroup of group $G$. Is it possible that a left coset of $H$ contains more than one right coset of $H$? It is clear to me that the answer is 'no' if we deal with finite groups.
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product of sets with finite complement in an infinite group [closed]

Let $G$ be an infinite group and $F\subseteq G$ be finite and $C=G\setminus F$. Is $CC$ equal to $G$?
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How to prove of non-existence of subgroup? [duplicate]

Let's get $A_4$ - it is group of even permutations. We have: $$|A_4| = 12$$. How to prove to following fact: $A_4$ do not contain subgroup $H$ such that $|H| = 6$
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Two collections of subsets of a group and their equality [closed]

Let $G$ be a group and $H\le G$ and let $$\mathcal T=\{B\subseteq G \mid (\forall b\in B)(bH\subseteq B)\}$$ and $$\mathcal S=\{B\subseteq G \mid (\forall b\in B)(Hb\subseteq B)\}$$ Is $\mathcal T$ ...
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$D_6$ is not a subset of $D_8$

I came across an example in Chapter-2 of Dummit and Foote(page-47) which says :$D_6$ is not a subgroup of $D_8$ ,the former is not even a subset of latter.I can't understand why is it not the subset ...
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$\mathbb S_n$ as semidirect product

In this note, I've read that $\mathbb S_n$ is a semidirect product of the alternating group $A_n$ by $\mathbb Z_2$. So I am trying to define a morphism $\rho: \mathbb Z_2 \to Aut(A_n)$ to show that ...
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Proving the existence of a subgroup.

Let $H$ be a subgroup of $\mathbb{Z}$. Prove that exists a subgroup $H'$ such that $H\cup H' = \mathbb{Z}$ and if $H$ is a non-trivial subgroup there is no group $H'$ such that $H \oplus H' = ...
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Semidirect product, normal subgroup exercise

Let $G$ be a group and let $H,K$ be subgroups of $G$ such that $G=H \rtimes K$. (i) Show that if $K \lhd G$, then $kh=hk$ for all $h \in H, k \in K$. (ii) Deduce that $G$ is abelian if and only if ...
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A number to the group cardinality power

Well my question is how is possible this: Consider an element $g\in G$, where $G$ is a finite group, then you have: $g^{|G|}=e$ How can I prove it? Thank you.
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Can a countable group have uncountably many distinct Hausdorff group topologies?

Question. Can a countable group have an uncountable number of distinct Hausdorff group topologies? By a group topology one understands a topology with respect to which the group operations are ...
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A question about holomorph of groups.

Question- If $G$ is complete, then the holomorph of $G$ is isomorphic to $G$×$G$. I am studying semidirect products for the first time, and in some notes i found this exercise. As far as i know about ...
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Point group of a disjoint union of graphs

Let $G$ be a graph. $\Gamma(G)$ is the point group og $G$, i.e. the automorphism group of $G$. Suppose $$G \cong nH $$ i.e. the disjoint union of $n$ graphs isomorphic to $H$. Then what is ...
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Infinite group having composition series

Example of an infinite group having composition series. I have examples as infinite alternating group and projective special linear group. But I want example other than infinite simple group.
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How can I construct a surjection with a kernel $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$?

How can I construct a surjection from a group $G$ to $S_3$ with the kernel $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2$ ? I know how to construct the surjection from $S_4$ to $S_3$ with ...