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

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Prove that every subgroup of an abelian group is a normal subgroup.

Prove that every subgroup of an abelian group is a normal subgroup. (Previously I had a solution but it was faulty towards the end of it. Sorry everyone, I was new to that "answer your own ...
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12 views

Finite groups of Mobius Transformations

Let $M_2(\mathbb{C})$ be the group of all Mobius transformations $z\mapsto \frac{az+b}{cz+d}$ from $\mathbb{C}\cup\{ \infty\}$ to itself. Let $PSU(2,\mathbb{C})$ be the group of all Mobius ...
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1answer
16 views

is the image of a group inside its profinite completion normal? characteristic?

Let $G$ be a finitely generated, residually finite group and $\widehat{G}$ its profinite completion. Must the natural image of $G$ inside $\widehat{G}$ be a normal subgroup of $\widehat{G}$? Must it ...
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1answer
28 views

Non-trivial group homomorphism from an infinite group to a finite group

Let $G$ be a topological group the underlying set of which is infinite (e.g., $(\mathbb{R}\,;+)$ or $(\mathbb{Z}\,;+)$), and let $H$ be a topological group the underlying set of which is finite (e.g., ...
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15 views

subsets of $\mathbb{Z}_2^{p}$ up to permutation equivalence

Let $\mathbb{Z}_2:= \mathbb{Z}/2\mathbb{Z}=\{0,1\}$. Let $p$ be a prime integer. We use $$\mathbb{Z}_2^{p}:= \mathbb{Z}_2\times \mathbb{Z}_2 \cdots \times\mathbb{Z}_2\qquad (p-times).$$ i.e., each ...
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126 views

Cosets/ Cyclic group

Let $G=\langle a\rangle$ and $H=\langle a^2\rangle$. Find all the right cosets of $H$ in $G$. Additional info: I understand that a right coset of $H$ in $G$ is of the form $Ha=\{ha:h \in H\}$. But I ...
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11 views

When does each action corresponds to a homomorphism and an anti-homomorphism.

If I adopt for function evaluation and function composition the convention $f(x)$ and $(f\circ g)(x) = f(g(x))$, and if $G$ is a right group action on some set $X$, then to this right action their ...
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Invariant factors of a subgroup of a subgroup of $\mathbb{Z}^2$

Consider groups $B\leq A\leq\mathbb{Z}^2$. We have: A basis $e_1,e_2$ of $\mathbb{Z}^2$ and integers $a_1,a_2$ such that $a_1e_1,a_2e_2$ is a basis for $A$, and $a_1\mid a_2$. A basis $f_1,f_2$ of ...
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11 views

Uniquely $p$-divisible group-Reference Request.

Define a group $G$ to be uniquely $p$-divisible if for all $x\in G$, there is a unique $y\in G$ such that $x=y^p$. Can someone kindly provide references where this class of groups is studied? Of ...
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43 views

$[U \cap H : G \cap H] \le [U :G]$

Is this always true that $[U \cap H : G \cap H] \le [U :G]$ , where U is a group and $G,H$ are subgroups of $U$? My trials for $\mathbb{Z}$ were giving affirmative answer but how to prove it, if it ...
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Does $U=U_1(\mathbb{Z}G)$ normalize $G$?

Let $G$ is an arbitrary group and and $U=U_1(\mathbb{Z}G)$ is the set of normalized units of $ZG$ i.e. $U_1(\mathbb{Z}G)$ here means units of $\mathbb{Z}G$ with augmentation $1$, i.e. set of all ...
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a question about abstract algebra,prove that the ring is commutative. [duplicate]

(1)A ring R is a booleean ring if for every $a\in R$,$a^2=a$. Show that every Boolean ring is a commutative ring. (2)Let R be a ring,where $a^3=a$ for all $a\in R$.Prove that R must be a commutative ...
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29 views

Equivalent presentation for the fundamental group of the projective plane

We know that $\langle a,b;(ab)^2=1\rangle$ and $\langle z;z^2\rangle$ are presentations of the fundamental group of the projective plane. Therefore, one is obtained from the other via Tietze ...
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1answer
47 views

Proving that the matrix exponential map is surjective onto the general linear group

Let $M_n(\mathbb{F})$ be the set of all $n\times n$ with entries in $\mathbb{F}$ and let $\exp:M_n(\mathbb{C})\to M_n(\mathbb{C})$ be defined by $$ \exp(A)=\sum_{k=0}^{\infty}\frac{A^k}{k!},$$ for ...
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1answer
36 views

Group Acting on a Ring

What would be the definition for a group action on a ring? I could not find one online. Would this be acceptable? A group action of a group G on a ring R is a map from G x R to R defined by g(r)=g.r ...
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1answer
33 views

Why this particular definition for the derivative on a group ring?

I am reading Introduction to Knot Theory by Crowell and Fox, and I am a bit confused at the way they define a derivative on a group ring (and on a group). I understand a derivative (or derivation ...
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1answer
22 views

Element order proof [on hold]

$n\in \mathbb{Z}$ and $\overline{a}\in U(\mathbb{Z}_n)$ order is $kl$. Prove that $\overline{a}^k$ order is $l$. Any ideas on how to approach this? It seems to follow straight from power definition. ...
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2answers
39 views

R is commutative ring with identity & define $\circ$ on $R$ by for any $a,b \in R$ $a \circ b=a+b-ab$ Prove the following

Let R be a commutative ring with identity. Define a new operation $\circ$ on $R$ by for any $a,b \in R$ $$a \circ b=a+b-ab$$ a) Prove that $\circ$ is associative b) Prove that R is a field iff the ...
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Is the profinite completion of $SL_2(\widehat{\mathbb{Z}})$ isomorphic to $\widehat{SL_2(\mathbb{Z})}$?

Is the profinite completion of $SL_2(\widehat{\mathbb{Z}})$ isomorphic to $\widehat{SL_2(\mathbb{Z})}$ ? Does anything change if we replace $SL$ with $GL$?
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40 views

Prove that $g\in \ker \varphi$

Let $G \approx \langle a,b \mid b^{-1}a^3b = a^5 \rangle$. Let $H$ be a finite group and $\varphi:G \to H$ be a homomorphism. Then $g = a^{-1}b^{-1} a^{-1}bab^{-1}ab \in \ker \varphi$. My attempt: ...
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Let R be a ring. Prove that if $x^2=x$ for each $x \in R$, then R is a commutative ring. [duplicate]

Let R be a ring. Prove that if $x^2=x$ for each $x \in R$, then R is a commutative ring. Ok, so I'm just looking for some confirmation that I'm doing this correctly. If we suppose $x,y \in R$ Let's ...
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1answer
17 views

Paradoxical Decomposition

A paradoxical decomposition of a group $G$ is a decomposition of $G$ into disjoint subsets: $G=U\cup S_1\cup S_2\cup\cdots\cup S_m\cup T_1\cup T_2\cup\cdots T_n$ so that there exist elements $g_1, ...
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1answer
30 views

Are these example subgroups?

Example 1 The subgroup $\{0,3\}$ of $\mathbb Z_6$ is a normal subgroup I know it's a subgroup because the element $3\in\mathbb Z_6$ generates $\{0,3\}$ under addition $\langle 3\rangle=(0,3)$ and ...
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2answers
104 views

Algorithm design for enumerating pairs of noncommuting elements up to conjugacy

I am trying to write some Magma code that, given a group $G$, returns a list of pairs $(x,y)$ with $x,y\in G$ such that $[x,y]\neq 1$ and such that every pair $(z,w)$ in the group with $[z,w]\neq 1$ ...
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1answer
34 views

understanding of Artin's proof of “$A_n$ is generated by $3$-cycles”

A quick proof for "$A_n$ is generated by $3$-cycles ($n\geq 4$)" is calculating the product of possible two $2$-cycles. I read the following different proof from Artin's Algebra(2nd): This is ...
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1answer
44 views

Subgroups of finite index have finitely many conjugates

is it true the following statement: Let $G$ be a group and let $H$ be a subgroup of $G$. If the index $[G:H]$ of $H$ in $G$ is finite, then $H$ have finitely many conjugates. What I think, is that ...
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Characteristic of a Finite Integral Domain

I am a little confused as how to approach this problem. The title of this problem is the title of the section which it comes from. However, there is no information that the given integral domain is ...
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How do I find the number of group homomorphisms from $S_3$ to $\mathbb{Z}/6\mathbb{Z}$? [duplicate]

How do I find the number of group homomorphisms from the symmetric group $S_3$ to $\mathbb{Z}/6\mathbb{Z}$?
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Is A4 isomorphic to D3 x Z2?

Prove/Disprove that $A_4\cong D_3×Z_2$: I can't seem to find a way to disprove it (Something in my mind it telling me it can't be true). But the order of both is 12, both are non cyclic, non ...
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Count the number group homomorphisms from $S_3$ to $\mathbb{Z}/6\mathbb{Z}$ ?

I have to count the number of group homomorphisms from $S_3$ to $\mathbb{Z}/6\mathbb{Z}$ ? 1 2 3 6 I aware of the formula for calculating group homomorphisms defined on cyclic groups..here ...
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Homomorphism or not?

Consider the function $$\phi :\begin{align} \mathbb Z_4 &\to \mathbb Z_4\\z &\mapsto 1\end{align}$$ Why is this not a group homomorphism? On the other hand Why $$\psi :\begin{align} ...
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1answer
31 views

Natural action of $\operatorname{Aut}(G)$ on sets of subgroups of $G$ of same order is transitive.

I am looking for the classification of those finite groups whose automorphism group acts transitively on sets of subgroups of same order. Let $G$ be a finite group and $d$ be a divisor of the order ...
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1answer
23 views

Homomorphisms preserve solubility of groups, and some others.

Definition: Let $G$ be a group. A subnormal series for G is a chain of subgroups $1 = G_0 \subseteq G_1 \subseteq G_2 \subseteq G_n = G$ such that $G_i$ is normal in $G_{i+1}$ for $i = 0,1, ..., ...
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1answer
60 views

Is it true that if $g,h\in G$ have order $p$, where $p$ is prime, the only possible order of $\langle g\rangle \cap \langle h \rangle$ is $p$?

Is it true that if $g,h\in G$ have order $p$, where $p$ is prime, the only possible order of $\langle g\rangle \cap \langle h \rangle$ is $p$? Here is my attempted proof of the result. Is it ...
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1answer
49 views

How to find $\left|\operatorname{Aut}(\mathbb Z_2\times\mathbb Z_2)\right|$ [duplicate]

How to find $\left|\operatorname{Aut}(\mathbb Z_2\times\mathbb Z_2)\right|=?$ Since $\mathbb Z_2\times\mathbb Z_2$ has 4 elements, $1 \leq \left|\operatorname{Aut}(\mathbb Z_2\times\mathbb ...
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2answers
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Why is the set of integers with the operation of addition considered a cyclic group?

The first sentence in the Wikipedia article entitled "Cyclic Groups" states that "In algebra, a cyclic group is a group that is generated by a single element". How is this consistent with addition on ...
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3answers
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Case when all subgroups of a group are normal

I was thinking in light of this question, where the groups G in discussion is of order 21 and question is to find number of non-trivial normal sub-groups of G. In the answers it is mentioned that ...
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1answer
42 views

Proving this homomorphism has a nontrivial kernel

Given: $|G|=n$, $H$ is a subgroup of $G$ and $|G/H|=k$, where $n$ does not divide k!. WTP: The left action map on $G/H$ has a nontrivial kernel. I have not put the entire problem I am trying to work ...
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1answer
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A doubt in a lemma on integral group rings.

In a paper by Farkas, I was doing this lemma, where I had this doubt (red underlined) in the proof of the lemma. Can anybody explain me how does it follow $\alpha$ is centralized by $H$. It should ...
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An Abstract Characterization of $S_5$ using involutions and their centralizers

This is essentially an exercise from Jacobson's Basic Algebra I. (p.83, ex.10) I've managed to solve all the other part of the proof, except (vi) and (x). I've been thinking about this all day, but ...
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Canonical Action of the Wreath Product on the Product Space.

This is essentially an exercise from Jacobson's Basic Algebra I. (p.79, ex.11) Exercise 11 says that the canonical action of the wreath product of $G$ and $H$ on the product space can be defined by ...
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1answer
35 views

Let g in G be an element of finite order. Find the order of $g^m$ where $m \ge 0$.

I understand how it goes from saying that md is a multiple of |g|. But am unable to understand why the statement right after is true.
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1answer
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left and right multiplication for Cayley graphs

Correct me if i am wrong but i have written down the following: If $X$ is a finite group, with subset $S$ and corresponding Cayley graph $G$ The edge set for a Cayley graph is defined such that two ...
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amalgamation of locally finite groups

It is well known that in category of groups there are Push-outs so it is possible to realize amalgamation in some kind of most free way. My question is what about category of locally free groups? I ...
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Consider group G acting on a set X

Consider group G acting on a set X Give examples of: a)The action that is transitive and faithful My Answer: Group G under addition acting on a set of integers Z b)The action that is transitive ...
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1answer
994 views

Irreducible representations of a cyclic group over a field of prime order

Consider $G$ a cyclic group of order $n$ with prime $p\nmid n$. How do I construct all the irreducible representations over $\mathbb F_p$? How many irreducible representations are there and what are ...
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1answer
50 views

Finite conjugate subgroup

In a paper titled "Trivial units in Group Rings" by Farkas, what does it mean by Finite conjugate subgroup. Here is the related image attached- What is finite conjugate subgroup of a group? It is ...
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1answer
48 views

Show that the free product of countably many countable groups is countable.

The question I am struggling with is as follows: Suppose that $\{G_\alpha\}$ is a countable collection of countable groups. Show that $\ast_{\alpha}G_\alpha$ is countable. The definition of ...
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1answer
28 views

Looking for examples of finite loops and monoids

I am looking for examples of (small) finite loops and monoids that are not groups for demonstrating what happens if you omit some of the group axioms. Does anyone know some ressources for this? I ...
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1answer
35 views

For a $p$-group $G$, and $H \le G$, if $G=H G^\prime$, then $H=G$ [on hold]

For a $p$-group $G$, and $H \le G$, prove that, if $G=H G^\prime$, then $H=G$; where $G^\prime$ is the commutator subgroup of $G$.