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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$S=\{a\in G:f(x)=f(ax)\; \forall x\in G\}$ is a subgroup of $G$

Let $G$ be a group and $f:G\rightarrow G$ a function. Let $S=\{a\in G:f(x)=f(ax)\; \forall x\in G\}$. Prove that $S$ is a subgroup of $G$. This is my first encounter with functions in this ...
2
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1answer
63 views

Is there a concise argument for why this group operation is associative?

Given disjoint groups $(G,\cdot)$ and $(H,\ast)$ and an isomorphism $f:G\to H$, I've been able to show that the binary operation $\diamond$ on $G\cup H$ defined by $a\diamond b = a\cdot b$ if $a,b ...
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1answer
97 views

Automorphism $f$ so that $f(x)=x^{-1}$ for half the members of the group: is it an involution?

Let $G$ a finite group. Let $f: G \to G$ an automorphism such that at least half the elements of the group are sent to their inverses, i.e $$\mathrm{card}(\{g \in G|f(g) = g^{-1}\}) \geq ...
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1answer
44 views

Show $C^\prime $ is a subgroup of $G$

Let $G$ be a group. Let $C^\prime =\{ a\in G:(ax)^2=(xa)^2 \;\forall x\in G \}$. Prove $C^\prime$ is a subgroup of $G$. I could easily show it for a similar problem but instead with $ax=xa$. I am ...
2
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1answer
54 views

Core of a subgroup and Norm

I have recently faced to a notion called the Norm of a group. What is the relasion between Norm and Core of a subgroup of a finite group? Can we say that for every subgroup, the Core is contained in ...
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1answer
80 views

Confusion about Actions of the Symmetric Group

I'm working on some practice questions and I am having trouble understanding actions of the symmetric group. I have the answers, but there were no explanations as to how they were derived. I feel ...
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1answer
59 views

Are there uncountably many distinct group operations on an uncountable set?

Call two group operations $\ast_1$ and $\ast_2$ on a set $S$ $distinct$ if there exist $s_1,s_2\in S$ such that $s_1\ast_1 s_2 \neq s_1 \ast_2 s_2$. I know that there are uncountably many distinct ...
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1answer
133 views

Automorphism of $\mathbb{Q}$

How can we show that any automorphism of $\mathbb{Q}$ under addition is of the form $x \to qx$ ,for some q in $\mathbb{Q}$. edited- I found the same question was asked by ...
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434 views

Does every group have a 'cyclization'?

Here's the question: Does every group have a 'cyclization'? That is, let $G$ be a group. Does there necessarily exist a cyclic group $C$ and a surjective homomorphism $\varphi:G\rightarrow C$ such ...
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115 views

Probability that an element belongs to an infinite subset of a set

$\newcommand{\Sym}{\operatorname{Sym}}$ I was studying $\Sym(\mathbb{N})$, the set consisting of all the bijections from $\mathbb{N}$ to itself. Since it is a group, the concept of "period of an ...
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37 views

Let $G$ be a finite group and let $a$ be an element of $G$. Then, $|cl ~(a)| = |G:C(a)|$

Let $G$ be a finite group and let $a$ be an element of $G$. Then, $|cl ~(a)| = |G:C(a)|$ where $cl ~(a)$ refers to the conjugacy class of $a$. The proof in the book which I am reading asks to prove ...
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0answers
38 views

Representation of a group, and finite index subspaces

Been working on this for a while and haven't gotten anywhere. I would really appreciate some hints. Let $G$ be a group, not necessarily finite. $V=\mathbb C [G] $ a vector space with basis $(e_g, ...
3
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3answers
109 views

When a group acts on a metric space isometrically, transitively, and faithfully, is it a closed subspace of the isometry group?

Suppose you have a group $G$ acting on $ (M,d)$ a compact metric space by isometries (meaning $d(gx,gy) = d(x,y)$ for all $x,y \in M$ and all $g \in G$), transitively and faithfully. You can define ...
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2answers
118 views

Category with only one object.

Suppose we have the category with only one object - the group G. Why can we think of the morphisms in this category as of the elements of the group G? I would be very grateful for explanation.
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0answers
43 views

Relation between $|H \lor K|$ , $|H|$ and $|K|$

Let $H$ and $K$ be subgroups of a finite group , then we know that the subgroup generated by $H \cup K$ i.e. $H \lor K$ is the smallest subgroup containing both $H$ and $K$ , then how can we relate ...
4
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1answer
138 views

Reduced Group algebras

Take a finite group and a field of characteristic zero. The group algebra is due to Maschke's theorem semisimple so that its a finite direct sum of matrix algebras over division algebras. I like to ...
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93 views

Groups with cyclic Commutator subgroup

Is anything known about class of groups with cyclic commutator subgroup?
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1answer
34 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 $S_2$?
2
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1answer
40 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 ...
2
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3answers
67 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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2answers
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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
62 views

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

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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1answer
87 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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35 views

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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1answer
85 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 ...
4
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1answer
84 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 $SO(n)$?
2
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1answer
81 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 ...
6
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4answers
425 views

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
76 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 ...
4
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1answer
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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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3answers
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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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1answer
63 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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1answer
41 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 :
4
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2answers
209 views

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 ...
6
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2answers
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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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1answer
59 views

$C\subseteq D$. Prove $\mathcal P (C)$ is a subgroup of $\mathcal P(D)$.

Let $C$ and $D$ be sets, with $C\subseteq D$. Prove $\mathcal P(C)$ is a subgroup of $\mathcal P(D)$. I can't easily see a proof for this, so I tried working on a counterexample. If I could just ...
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1answer
77 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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1answer
90 views

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 ...
3
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2answers
102 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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1answer
63 views

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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2answers
67 views

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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1answer
91 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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3answers
120 views

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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3answers
105 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.
5
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2answers
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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 $p$ ...
2
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2answers
54 views

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 ...