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

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Reduced Group algebras

Take a finite group and a field of characteristic zero. The group algebra is due to Maschkes theorem semisimple so that its a finite direct sum of matrix algebras over division algebras. I like to ...
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Groups with cyclic Commutator subgroup

Is anything known about class of groups with cyclic commutator subgroup?
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25 views

Onto homorphisms from $S_4$ to $S_2$

Let $S_n$ represent the symmetric group on $n$ letters. How can one find an onto homomorphism from $S_4$ to $S_2$?
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1answer
18 views

Injectivity of Natural Homomorphism to Groupification

This is a continuation of my own question some time ago. Suppose $M$ is a monoid and $G$ is the groupification of $M$. (I figure groupification of $M$ is a better term than Grothendieck group of ...
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31 views

Isomorphism of Group Products

Let $G$ be a group, $A = G \times G$. In $A$, Let $T = \{(g, g)|g \in G\}$. Prove that $T$ is isomorphic to $G$. I don't know how to continue this problem. $A$ is abelian. Therefore, $G \times G$ is ...
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every group over the sets of a partition of a goup is well defined.

I would like help with the following problem in Dummit and Foote: Let $P$ be a partition of the elements of the group $G$. we define the set consisting of the sets of the partition $P$ is a group ...
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1answer
44 views

Is there an abelian group $G$ such that $|G| > 333$ and $x^3=e$ for all $x \in G$ [on hold]

Is there an abelian group $G$ such that the order of $G$ is greater than $333$ and $x^3=e$ for all $x \in G,$ given $e$ is the identity. If the answer is yes, please give an example.
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65 views

Are any two groups of order 23 isomorphic to each other?

I have to decide whether the following statements are true or false, with proofs. Any two abelian groups of order $23$ are isomorphic to each other Any two abelian groups of order $25$ are ...
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finite group homology: $nH_k(G;M)=0$ for $n=|G|$?

Let $G$ be a finite group. Is there a simple proof (if any) that the order of $G$ annihilates the Eilenberg-MacLane homology $H_k(G;M)$ for all $k\geq1$? A simple proof of the statement for ...
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30 views

Is it possible to represent subsets of natural numbers as groups with prime generators?

I'm learning group theory and I'm trying to consider the "symmetry" of a certain group of natural numbers: Here's the idea, all natural numbers are comprised of multiples of primes. So a subset would ...
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22 views

Same number of independent parameters for $SO(n)$ and $O(n)$

Why is the number of independent parameters for $SO(n)$ and $O(n)$ same, in spite of an additional constraint of unit determinant for $SO(n)$?
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45 views

Exercise on characterization of free abelian groups

I was wondering if someone can please check my work on a homework problem. This is from the graduate Hungerford text. Chapter 2.1, number 3. Let $X=\{a_i\ |\ i\in I\}$ be a set. Then the free abelian ...
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There are finitely many maps on nonnegative integers satisfying $\phi(ab)=\phi(a)+\phi(b)$

How to show that there are finitely many maps $\phi:\mathbb{N}\cup\{0\}\to \mathbb{N}\cup\{0\}$ with the property that $\phi(ab)=\phi(a)+\phi(b)$ for all $a,b\in \mathbb{N}\cup\{0\}$.
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1answer
31 views

Prove a result on transitive group actions.

Let $G$ be a group and $A$ & $B$ be two sets s.t. $G$ acts transitively on each of $A$ & $B$. Choose some $\alpha$ and $\beta$ in $A$ & $B$ respectively then prove that if $G=G_\alpha ...
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Defining a subgroup of $GL(2,7)$ in GAP

Considering this resent post in which $|G|=42$, I am thinking of making this subgroup concrete in GAP environment. Maybe, if the structure of $G$ was known then, we would use an appropriate mapping ...
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Subgroup of matrices exercise

Let $G=\{\left( \begin{array}{ccc}1 & b \\ 0 & a \\ \end{array} \right) : a,b \in \mathbb Z_7, a \neq 0\}$. Find the order of $G$. For each prime $p$ such that $p$ divides $|G|$, find all ...
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41 views

Equality of subgroups $H \subseteq K \subseteq G$ with the same finite index in $G$

Let $G$ be a group and let $H,K$ be two subgroups of $G$ such that $H\subseteq K$and $[G:H] = [G:K]$ is finite. Prove that $H = K$. Can somebody please give me some idea to solve this?
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23 views

Order of an element in direct product using cayley's diagram

How can I find the order of element (1,1) of the group $C_4\times C_3$ visually in the diagram below :
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Let $S$ be a subset of a group $G$ that contains the identity element $1$ and such that the left cosets $aS$ with $a$ in $G$, partition $G$.

Let $S$ be a subset of a group $G$ that contains the identity element $1$ and such that the left cosets $aS$ with $a$ in $G$, partition $G$.Prove that $S$ a is a subgroup of $G$. My try: For $h$ in ...
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126 views

Existence of a minimal generator for a group?

Let $G$ be a group and $A\subseteq G$ and $G=\left<A\right>$. Is there a minimal $B\subseteq A$ with $G=\left<B\right>$?
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Is there a theory of “derived extensions”?

Given an exact sequence of groups $$1\rightarrow N\rightarrow G\rightarrow K\rightarrow 1$$ we call $G$ a central extension of $K$ by $N$ if the image of $N$ is contained in the center of $G$. Central ...
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29 views

$C\subseteq D$. Prove $P_C$ is a subgroup of $P_D$.

Let $C$ and $D$ be sets, with $C\subseteq D$. Prove $P_C$ is a subgroup of $P_D$. I can't easily see a proof for this, so I tried working on a counterexample. If I could just show that $P_C$ is ...
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46 views

$H$ not closed under addition due to inverses, but closed under inverses

I have a fairly basic question. Problem from my text: $G=\left \langle \mathbb{R}^2 ,+\right\rangle, H=\{(x,y):x^2+y^2>0\}. $ Determine whether H is a subgroup of G. It's easy to show that ...
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Understanding Dynkin Diagrams - any organising ideas - are they now adequately understood?

Some 30 or so years ago JH Conway posed a question about the ubiquity of the Dynkin Diagram - not necessarily in public, but I heard him ask it. I think it was in the context of "what would be ...
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55 views

Subgroup Decision Problem

Let $x \in G$ be an element of a group $G$ of order $n = pq$. $G_p$ and $G_q$ are prime order sub-groups of order $p$ and $q$ respectively. How can we prove that $x^q \in G_p$? I want to understand ...
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order of an elliptic curve

I have found that the curve given by $x^3+x+1=y^2$ over $\mathbb{F_5}$ has 9 points. Now I am supposed to find the number of points of the same curve on $\mathbb{F}_{125}$. Using Hasse and the fact ...
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If $x^2 a x=a^{-1}$, then $a$ has a cube root. [duplicate]

In a group $G$: If $x^2 a x=a^{-1}$, then $a$ has a cube root. (Hint: Show that $xax$ is a cube root of $a^{-1}$.) So essentially $\exists y\in G:a=y^3$. The hint probably confused me more than ...
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If $a^3=e$, then $a$ has a square root.

Assuming $a\in G$ where $G$ is a group. I'm not sure why this is hard for me. Essentially, the problem is just saying: If $a^3=e$, then $\exists x \in G : a=x^2$. Can somebody give me a hint or a ...
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Finding complements of direct summands

Let $B=\mathbb Z⊕\mathbb Z_4$. How could we prove that $B_1=(1,\bar 1)\mathbb Z$ and $B_2=(1,\bar 2)\mathbb Z$ are direct summands in $B$? Or, the same question for $A=\mathbb Z⊕\mathbb Z$ and ...
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53 views

Is infinite product of groups also a group? [on hold]

I've been seacrhing the answer and I cant find it. I would be glad if you give me reference to read. Thanks.
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Is Hom$(G_1, G_2)$ a group?

The collection of all homomorphisms from the group $G_1$ to the group $G_2$ is denoted as Hom$(G_1, G_2)$. I am willing to show that if $G_1 \simeq G_1'$ then Hom$(G_1, G_2) \simeq$ Hom$(G1', G2)$. ...
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66 views

Are there at least denumerably many distinct group operations on any denumerable set?

I'm working on a proof of the following statement: For any denumerable set $D$, there exist at least denumerably many distinct group operations on $D$. My argument is looking fairly messy, so I'm ...
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Group of even order must contain $a:a=a^{-1}$ $ (a\not = e)$ [duplicate]

Let $G$ be a finite group. If the order of $G$ is even, prove that there is at least one element $a$ in $G$ such that $a\not= e$ and $a=a^{-1}$. Here's my idea: Suppose $\{x_1,\cdots,x_n\}$ is ...
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70 views

An epimorphism from $\mathbb Z⊕\mathbb Z⊕\cdots$ to $\mathbb Q$

I want an explicit example of an epimorphism from $\mathbb Z⊕\mathbb Z⊕\cdots$ to $\mathbb Q$. Thanks.
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If $p\mid|G|$ then how many elements of order $p$ are there in $G$?

Let $G$ be a finite group and $p$ be a prime such that $p\mid|G|$ , then obviously $G$ has an element of order $p$ (by Cauchy's theorem) ; I would like to know exactly how many elements of order $p$ ...
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Show $H_n$ is a subgroup of $G$ for any $n\in\mathbb N$

Let $G = GL_2(\mathbb C)$ be the group of invertible complex $2 \times 2$ matrices and for each $n\in\mathbb N$ consider the subset: $$H_n = \{\,A \in G : (\det A)^n= 1\,\}.$$ I know to prove if it ...
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Weil restriction for schemes

I'm trying to understand the Weil restriction of a scheme (since I'm reading a paper which uses it). I'm even having troubles trying understanding the following "toy" example. Toy example. Let $X$ be ...
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Question about particular words in the free group on three generators

In the free group generated by the letters $x,y,z$ suppose that we have a word such that for any one of $x,y,z$ the indices of each occurrence of that letter in our word sum to zero. Suppose further ...
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Constructing a group with given normal subgroups.

Let $N_1,N_2,\dots,N_n$ be simple groups. Is there is a group $G$ with exactly $n$ nontrivial proper normal subgroups isomorphic to $N_1,N_2,\dots,N_n$?
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80 views

How does $A_n$ look in Aut$(X)$?

Let me phrase my question precisely: Let $X=\{1,2,3,...,n\}$, $ \ S_n=\mbox{Sym}\{1,2,3,...,n\}$ be symmetric group on $n-$letters. Let $\mbox{Aut}(X)$ denote the automorphism group of $X$. We ...
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Generators of the Symmetric group $S_3$

I am trying to find the generator(s) of the Symmetric Group $S_3$ and I have attempted this via brute force by listing the permutations of $S_3$ and composing and repeating them but I have not found ...
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A Group Isomorphic To The Direct Product Of Two of its Subgroups. Then are These Subgroups Normal?

Let $G_1$ and $G_2$ be subgroups of a group $G$. Assume that $G$ is isomorphic to $G_1\times G_2$. Then is it necessary that $G_1$ and $G_2$ are normal in $G$? Clealry $G_1\cong ...
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In a group of exponent $2^n$, $[x^{2^{n-1}},y^{2^{n-1}},\ldots,y^{2^{n-1}}]=1$?

In a group of exponent $2^n$, is the following equality true? $[x^{2^{n-1}},\underbrace{y^{2^{n-1}},\ldots,y^{2^{n-1}}}_n]=1$. Here, $[a, b, c]=[[a, b], c]$. Call the above question "Question ...
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Finite Linearly Ordered Abelian Monoids

This question concerns the proper definition of the phrase "finite linearly ordered abelian monoids". The sequence A030453 of OEIS counts the number of "finite linearly ordered abelian monoids". The ...
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Variational formulations in group theory?

I apologise if this is a naïve question. Are there any known / widely applicable / important variational formulations in (finite) group theory? That is, a relationship of the form $$\alpha(G) = ...
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1answer
54 views

Showing that the Binary Icosahedral Group (given by a presentation) has order $120$.

Say we have a group generated by $a, b$, with the relations $(ab)^2=a^3=b^5$ (note that these are not necessarily equal to the identity). How do I show that the group has $120$ elements? Without ...
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What does $\frac12(D_{2p}\times D_{2p})$ mean in group theory?

Reading a thesis, I have come across the (unexplained) notation $$\frac{1}{2}(D_{2p}\times D_{2p})\cong (p\times p):2,$$ where $D_{2p}$ is a dihedral group. What does this "$\frac12$" notation mean? ...
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141 views

Normal Groups and Quotient Groups

These concepts are currently confusing me. My reading first defined a normal subgroup as one that is the kernel of a group homomorphism. Then it introduced the terms "left coset" and "right coset," ...
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Normal subgroups, direct product and monomorphism problem

Let $G$ be a group and let$H,K$ be normal subgroups of $G$. Let $\pi_H,\pi_K$ be the projections on $H$ and $K$ respectively. Show that the map $$f:G/(H \cap K) \to G/H \times G/K$$ defined as ...
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Counting of edges coloring in a graph

The problem is to count of coloring graphs. We have three colors. And I found all automorphisms. It is: $$\alpha_1: (1)(2)(3)(4)(5)(6)$$ $$\alpha_2: (123456) $$ $$\alpha_3: (135)(246) $$ ...